1. Introduction

In differential calculus, we learned how to find the derivative of a function. Now we ask the reverse question: given a function f(x), can we find a function F(x) whose derivative is f(x)? This reverse process is called integration, and the function F(x) is called an antiderivative or primitive function of f(x).

The study of antiderivatives forms the foundation of integral calculus and has profound applications in physics, engineering, economics, and many other fields. For example:

If velocity v(t) is known, finding position requires an antiderivative
If acceleration a(t) is known, finding velocity requires an antiderivative
If marginal cost is known, finding total cost requires an antiderivative

2. Definition of Antiderivative

Definition 5.1: Antiderivative (Primitive Function)

Let $f(x)$ be a function defined on an interval $I$ . A function $F(x)$ is called an antiderivative (or primitive function) of $f(x)$ on $I$ if:

F'(x) = f(x) \quad \text{for all } x \in I

Remark 5.1: Terminology

The terms “antiderivative” and “primitive function” are used interchangeably. “Primitive” is more common in European literature.

Example 5.1: Basic Antiderivatives

Problem: Find an antiderivative of each:

(a) $f(x) = x^2$ (b) $f(x) = \cos x$ (c) $f(x) = e^x$

Solution:

(a) Since $\frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2$ , an antiderivative is $F(x) = \frac{x^3}{3}$ .

(b) Since $\frac{d}{dx}(\sin x) = \cos x$ , an antiderivative is $F(x) = \sin x$ .

(c) Since $\frac{d}{dx}(e^x) = e^x$ , an antiderivative is $F(x) = e^x$ .

Theorem 5.1: Uniqueness up to Constant

If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ on interval $I$ , then there exists a constant $C$ such that:

G(x) = F(x) + C \quad \text{for all } x \in I

Proof:

Given: $F'(x) = f(x)$ and $G'(x) = f(x)$ for all $x \in I$ .

To Show: $G(x) - F(x) = C$ for some constant.

Proof: Let $H(x) = G(x) - F(x)$ . Then:

H'(x) = G'(x) - F'(x) = f(x) - f(x) = 0 \quad \text{for all } x \in I

By MVT corollary, if $H'(x) = 0$ on an interval, then $H(x)$ is constant. Therefore $G(x) - F(x) = C$ . ∎

∎

Corollary 5.1: General Form of Antiderivatives

If $F(x)$ is any antiderivative of $f(x)$ , then the general antiderivative is $F(x) + C$ where $C$ is an arbitrary constant.

3. Indefinite Integral Notation

Definition 5.2: Indefinite Integral

If $f(x)$ has an antiderivative $F(x)$ on interval $I$ , then the indefinite integral of $f(x)$ is:

\int f(x)\,dx = F(x) + C

where: $\int$ is the integral sign, $f(x)$ is the integrand, $x$ is the variable, $dx$ is the differential, $C$ is the constant of integration.

Remark 5.2: Integral Symbol

The integral sign ∫ is an elongated “S” (for “summa” in Latin). It was introduced by Leibniz in 1675.

Example 5.2: Writing Indefinite Integrals

(a) $\int x^2\,dx = \frac{x^3}{3} + C$

(b) $\int \cos x\,dx = \sin x + C$

(c) $\int \frac{1}{x}\,dx = \ln |x| + C$

4. Fundamental Properties

Theorem 5.2: Inverse Relationship

If $f(x)$ has an antiderivative, then:

\frac{d}{dx}\left(\int f(x)\,dx\right) = f(x)

and if $F(x)$ is differentiable:

\int F'(x)\,dx = F(x) + C

Proof:

For the first: Let $\int f(x)\,dx = F(x) + C$ where $F'(x) = f(x)$ . Then:

\frac{d}{dx}\left(\int f(x)\,dx\right) = \frac{d}{dx}(F(x) + C) = F'(x) = f(x)

For the second: Since $F(x)$ is an antiderivative of $F'(x)$ , $\int F'(x)\,dx = F(x) + C$ . ∎

∎

Remark 5.3: Inverse Operations

Differentiation and integration are inverse operations, but not perfect inverses: $\frac{d}{dx}\int f\,dx = f$ but $\int \frac{d}{dx}f\,dx = f + C$ .

Theorem 5.3: Linearity of Indefinite Integrals

If $f(x)$ and $g(x)$ have antiderivatives, then:

(1) Sum Rule:

\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx

(2) Constant Multiple Rule:

\int kf(x)\,dx = k\int f(x)\,dx

Example 5.3: Using Linearity

Problem: Evaluate $\int (3x^2 - 5\cos x + 2)\,dx$

Solution:

\int (3x^2 - 5\cos x + 2)\,dx = 3\int x^2\,dx - 5\int \cos x\,dx + 2\int 1\,dx

= 3 \cdot \frac{x^3}{3} - 5\sin x + 2x + C = x^3 - 5\sin x + 2x + C

5. Existence of Antiderivatives

Theorem 5.4: Existence for Continuous Functions

If $f(x)$ is continuous on an interval $I$ , then $f(x)$ has an antiderivative on $I$ .

Remark 5.4: Proof Preview

The proof relies on the Fundamental Theorem of Calculus (Chapter 6): $F(x) = \int_a^x f(t)\,dt$ is an antiderivative of $f(x)$ .

Theorem 5.5: Darboux Property

If $F(x)$ is differentiable on $[a,b]$ , then $F'(x)$ satisfies the intermediate value property.

Corollary 5.2: Functions Without Antiderivatives

A function with a jump discontinuity cannot have an antiderivative on any interval containing the jump.

Example 5.4: Function Without Antiderivative

Problem: Show sgn(x) has no antiderivative on any interval containing 0.

Solution:

The sign function has jump discontinuity at x = 0:

\lim_{x \to 0^-} \text{sgn}(x) = -1 \neq 1 = \lim_{x \to 0^+} \text{sgn}(x)

By Corollary 5.2, it cannot have an antiderivative on any interval containing 0.

6. More Worked Examples

Example 5.5: Verifying an Antiderivative

Problem: Verify $F(x) = x\ln x - x$ is an antiderivative of $f(x) = \ln x$ .

Solution:

F'(x) = \frac{d}{dx}(x\ln x - x) = \ln x + x \cdot \frac{1}{x} - 1 = \ln x + 1 - 1 = \ln x

Since $F'(x) = f(x)$ , verified. ✓

Example 5.6: Antiderivative of |x|

Problem: Find $\int |x|\,dx$

Solution:

For $x \geq 0$ : $|x| = x$ , antiderivative is $x^2/2$ .

For $x < 0$ : $|x| = -x$ , antiderivative is $-x^2/2$ .

\int |x|\,dx = \frac{x|x|}{2} + C

Example 5.7: Simplify Then Integrate

Problem: Evaluate $\int \frac{x^2 + 1}{x^2}\,dx$

Solution:

\frac{x^2 + 1}{x^2} = 1 + \frac{1}{x^2} = 1 + x^{-2}

\int \left(1 + x^{-2}\right)\,dx = x - \frac{1}{x} + C

Example 5.8: Trigonometric Identity

Problem: Evaluate $\int \tan^2 x\,dx$

Solution:

Use identity $\tan^2 x = \sec^2 x - 1$ :

\int \tan^2 x\,dx = \int (\sec^2 x - 1)\,dx = \tan x - x + C

Example 5.9: Polynomial

Problem: Evaluate $\int (4x^3 - 6x^2 + 2x - 5)\,dx$

Solution:

\int (4x^3 - 6x^2 + 2x - 5)\,dx = x^4 - 2x^3 + x^2 - 5x + C

Example 5.10: Radical Function

Problem: Evaluate $\int \sqrt{x}\,dx$

Solution:

\int \sqrt{x}\,dx = \int x^{1/2}\,dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x\sqrt{x} + C

Example 5.11: Reciprocal Power

Problem: Evaluate $\int \frac{1}{x^3}\,dx$

Solution:

\int x^{-3}\,dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C

Example 5.12: Sum of Trig Functions

Problem: Evaluate $\int (\sin x + \cos x)\,dx$

Solution:

\int (\sin x + \cos x)\,dx = -\cos x + \sin x + C

Antiderivatives Quiz

12

Questions

0

Correct

0%

Accuracy

1

If

F'(x) = f(x)

for all

x \in I

, then

F(x)

is called:

Easy

Not attempted

2

If

F(x)

and

G(x)

are both antiderivatives of

f(x)

, then:

Easy

Not attempted

3

The symbol

\int f(x)\,dx

represents:

Easy

Not attempted

4

(\int f(x)\,dx)' = f(x)

shows that:

Medium

Not attempted

5

\int f'(x)\,dx = f(x) + C

shows that:

Medium

Not attempted

6

\int [f(x) + g(x)]\,dx =

Medium

Not attempted

7

The function

f(x) = |x|

on

\mathbb{R}

:

Hard

Not attempted

8

Which has no elementary antiderivative?

Hard

Not attempted

9

If

f

is continuous on

[a,b]

, then:

Medium

Not attempted

10

\int 0\,dx =

Easy

Not attempted

11

\int kf(x)\,dx =

Easy

Not attempted

12

\frac{d}{dx}\int f(x)\,dx = f(x)

holds when:

Medium

Not attempted

Frequently Asked Questions

What is the difference between antiderivative and indefinite integral?

An antiderivative F(x) is a specific function with F'(x) = f(x). The indefinite integral ∫f(x)dx is the family of all antiderivatives: F(x) + C.

Why do we add '+C' to indefinite integrals?

Because if F(x) is an antiderivative, so is F(x) + C for any constant. The '+C' represents all possible antiderivatives.

Does every function have an antiderivative?

No. But continuous functions always have antiderivatives (Fundamental Theorem). Functions with jump discontinuities may not.

What is the geometric meaning of an antiderivative?

If F'(x) = f(x), then F(x) has slope f(x) at each point. The graph of F has tangent lines with slopes given by f.

Can antiderivatives be non-elementary?

Yes! Functions like e^{-x²}, sin(x)/x have antiderivatives that cannot be expressed with elementary functions.

What is a primitive function?

Same as antiderivative. If F'(x) = f(x), F is a primitive of f. 'Primitive' is more common in European texts.

How do I verify F(x) is an antiderivative of f(x)?

Differentiate F(x) and check if F'(x) = f(x). E.g., d/dx(x³/3) = x² verifies x³/3 is antiderivative of x².

Why is it called 'indefinite' integral?

It has no specific limits (unlike definite integral ∫_a^b). The result is a function + constant, not a number.

What is the relationship between ∫ and d/dx?

They are inverse: d/dx[∫f dx] = f and ∫[f' dx] = f + C. Integration adds an arbitrary constant.

Can discontinuous functions have antiderivatives?

Sometimes. Removable discontinuities: yes. Jump discontinuities: no (derivatives satisfy IVP).

7. Common Mistakes to Avoid

Mistake 1: Forgetting +C

Wrong: $\int x^2\,dx = x^3/3$

Right: $\int x^2\,dx = x^3/3 + C$

Mistake 2: Wrong Power Rule

Wrong: $\int x^n\,dx = nx^{n-1} + C$

Right: $\int x^n\,dx = x^{n+1}/(n+1) + C$ for n ≠ -1

Mistake 3: Integrating 1/x

Wrong: $\int 1/x\,dx = \ln x + C$

Right: $\int 1/x\,dx = \ln |x| + C$

Mistake 4: Sign Error in Trig

Wrong: $\int \sin x\,dx = \cos x + C$

Right: $\int \sin x\,dx = -\cos x + C$

Mistake 5: Multiple Constants

Wrong: $\int(f+g)dx = F + C_1 + G + C_2$

Right: $\int(f+g)dx = F + G + C$ (one constant)

8. Study Tips

1. Think Backwards

Ask: “What function has this as its derivative?”

2. Always Verify

Check by differentiating: if $F'(x) = f(x)$ , correct!

3. Memorize Basics

Know: $x^n, e^x, \sin x, \cos x, 1/x, 1/(1+x^2)$

4. Never Forget +C

The constant represents all possible antiderivatives.

5. Simplify First

Expand products, simplify fractions before integrating.

6. Use Linearity

Break complex integrals into simpler pieces.

9. Quick Reference

Key Formulas

Definition:

F'(x) = f(x) \Rightarrow F \text{ is antiderivative of } f

Indefinite Integral:

\int f(x)\,dx = F(x) + C

Inverse Property 1:

\frac{d}{dx}\int f(x)\,dx = f(x)

Inverse Property 2:

\int F'(x)\,dx = F(x) + C

Sum Rule:

\int [f + g]\,dx = \int f\,dx + \int g\,dx

Constant Multiple:

\int kf(x)\,dx = k\int f(x)\,dx

10. Additional Practice Examples

Example 5.13: Exponential Function

Problem: Evaluate $\int 5e^x\,dx$

Solution:

\int 5e^x\,dx = 5\int e^x\,dx = 5e^x + C

Example 5.14: Secant Squared

Problem: Evaluate $\int \sec^2 x\,dx$

Solution:

Since $\frac{d}{dx}(\tan x) = \sec^2 x$ :

\int \sec^2 x\,dx = \tan x + C

Example 5.15: Cosecant Squared

Problem: Evaluate $\int \csc^2 x\,dx$

Solution:

Since $\frac{d}{dx}(-\cot x) = \csc^2 x$ :

\int \csc^2 x\,dx = -\cot x + C

Example 5.16: Inverse Trig Integral

Problem: Evaluate $\int \frac{1}{1+x^2}\,dx$

Solution:

Since $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$ :

\int \frac{1}{1+x^2}\,dx = \arctan x + C

Example 5.17: Another Inverse Trig

Problem: Evaluate $\int \frac{1}{\sqrt{1-x^2}}\,dx$

Solution:

Since $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ :

\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C

Example 5.18: Combined Powers

Problem: Evaluate $\int (x^4 - 3x^2 + 7x - 2)\,dx$

Solution:

\int (x^4 - 3x^2 + 7x - 2)\,dx = \frac{x^5}{5} - x^3 + \frac{7x^2}{2} - 2x + C

Example 5.19: Fractional Exponent

Problem: Evaluate $\int x^{-1/2}\,dx$

Solution:

\int x^{-1/2}\,dx = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C

Example 5.20: Mixed Expression

Problem: Evaluate $\int \left(\frac{1}{x} + \frac{1}{x^2}\right)\,dx$

Solution:

\int \left(\frac{1}{x} + \frac{1}{x^2}\right)\,dx = \ln|x| - \frac{1}{x} + C

11. Challenge Problems

Challenge 1

Find the antiderivative:

\int \frac{x^3 - 2x + 1}{x^2}\,dx

Show hint

Divide each term by x². Result: x - 2/x - 1/x + C

Challenge 2

Find the antiderivative:

\int (\sqrt{x} + \sqrt[3]{x})\,dx

Show hint

Write as x^{1/2} + x^{1/3}, then use power rule.

Challenge 3

Evaluate:

\int \sec x \tan x\,dx

Show hint

Recall d/dx(sec x) = sec x tan x. Answer: sec x + C

Challenge 4

Prove that if F is an antiderivative of f on (a,b) and G is an antiderivative of f on (b,c), we cannot always combine them into an antiderivative on (a,c).

Show hint

Consider continuity at b. The combined function may have a jump.

12. Geometric Interpretation

Slope Fields

If $F'(x) = f(x)$ , then at each point $(x, F(x))$ on the graph of $F$ , the slope of the tangent line equals $f(x)$ . The family of antiderivatives $F(x) + C$ gives parallel curves, each with the same slope at any given x-value.

This is why a slope field (or direction field) for $y' = f(x)$ shows the same slope pattern at all heights for a fixed x. All solution curves are vertical translates of each other.

Initial Value Problems

To find a unique antiderivative, we need an initial condition. Given $F'(x) = f(x)$ and $F(a) = b$ , there is exactly one antiderivative satisfying both conditions.

This picks out one specific curve from the family $F(x) + C$ by requiring it to pass through the point $(a, b)$ .

Example 5.21: Initial Value Problem

Problem: Find $F(x)$ if $F'(x) = 2x$ and $F(1) = 3$ .

Solution:

First, find the general antiderivative:

F(x) = \int 2x\,dx = x^2 + C

Apply the initial condition $F(1) = 3$ :

F(1) = 1^2 + C = 3 \Rightarrow C = 2

Therefore, $F(x) = x^2 + 2$ .

Example 5.22: Physics Application

Problem: A particle moves with velocity $v(t) = 3t^2 - 2t$ . If its position at t=0 is s(0)=5, find s(t).

Solution:

Since velocity is the derivative of position:

s(t) = \int v(t)\,dt = \int (3t^2 - 2t)\,dt = t^3 - t^2 + C

Apply $s(0) = 5$ :

s(0) = 0 - 0 + C = 5 \Rightarrow C = 5

Therefore, $s(t) = t^3 - t^2 + 5$ .

13. Historical Context

The concept of antiderivatives emerged in the 17th century through the work of Isaac Newton and Gottfried Wilhelm Leibniz, who independently developed calculus. Newton called antiderivatives “fluents” while Leibniz introduced the integral notation ∫ that we use today.

The profound discovery was that finding areas under curves (integration) and finding tangent slopes (differentiation) are inverse operations—the Fundamental Theorem of Calculus.

The term “primitive function” comes from the French “fonction primitive” and is still preferred in many European countries. The notation $\int f(x)\,dx$ was Leibniz's contribution, with the elongated S representing “summa” (sum).

14. Summary Table

Function f(x)	Antiderivative F(x)	Domain
$x^n$ (n ≠ -1)	$\frac{x^{n+1}}{n+1} + C$	Depends on n
$\frac{1}{x}$	$\ln\|x\| + C$	x ≠ 0
$e^x$	$e^x + C$	$\mathbb{R}$
$\sin x$	$-\cos x + C$	$\mathbb{R}$
$\cos x$	$\sin x + C$	$\mathbb{R}$
$\sec^2 x$	$\tan x + C$	x ≠ π/2 + nπ
$\csc^2 x$	$-\cot x + C$	x ≠ nπ
$\frac{1}{1+x^2}$	$\arctan x + C$	$\mathbb{R}$
$\frac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$	\|x\| < 1

15. More Initial Value Problems

Example 5.23: Acceleration to Position

Problem: Given acceleration $a(t) = 6t$ , $v(0) = 2$ , and $s(0) = 1$ , find position s(t).

Solution:

First find velocity:

v(t) = \int 6t\,dt = 3t^2 + C_1

Using $v(0) = 2$ : $C_1 = 2$ , so $v(t) = 3t^2 + 2$

Now find position:

s(t) = \int (3t^2 + 2)\,dt = t^3 + 2t + C_2

Using $s(0) = 1$ : $C_2 = 1$

Therefore, $s(t) = t^3 + 2t + 1$ .

Example 5.24: Economics Application

Problem: Marginal cost is $C'(x) = 20 + 0.1x$ . If fixed costs are $500, find total cost C(x).

Solution:

C(x) = \int (20 + 0.1x)\,dx = 20x + 0.05x^2 + C

Fixed costs mean $C(0) = 500$ :

C(0) = 0 + 0 + C = 500 \Rightarrow C = 500

Therefore, $C(x) = 20x + 0.05x^2 + 500$ .

Example 5.25: Population Growth

Problem: Population growth rate is $P'(t) = 100$ (constant). If $P(0) = 5000$ , find P(t).

Solution:

P(t) = \int 100\,dt = 100t + C

Using $P(0) = 5000$ : $C = 5000$

Therefore, $P(t) = 100t + 5000$ .

16. Extended Formula Table

Function	Antiderivative
$a^x$ (a > 0, a ≠ 1)	$\frac{a^x}{\ln a} + C$
$\sec x \tan x$	$\sec x + C$
$\csc x \cot x$	$-\csc x + C$
$\sinh x$	$\cosh x + C$
$\cosh x$	$\sinh x + C$
$\frac{1}{x^2+a^2}$	$\frac{1}{a}\arctan\frac{x}{a} + C$
$\frac{1}{\sqrt{a^2-x^2}}$	$\arcsin\frac{x}{a} + C$
$\frac{1}{x^2-a^2}$	$\frac{1}{2a}\ln\left\|\frac{x-a}{x+a}\right\| + C$
$\frac{1}{\sqrt{x^2 \pm a^2}}$	$\ln\|x + \sqrt{x^2 \pm a^2}\| + C$

17. Differentiation vs Integration

Aspect	Differentiation	Integration
Definition	Limit of difference quotient	Reverse of differentiation
Result	Unique function	Family of functions (+ C)
Power rule	$nx^{n-1}$	$\frac{x^{n+1}}{n+1}$
sin x	$\cos x$	$-\cos x$
cos x	$-\sin x$	$\sin x$
e^x	$e^x$	$e^x$
ln x	$1/x$	$x\ln x - x$
Existence	Not always exists	Exists if f continuous

18. More Advanced Examples

Example 5.26: Exponential with Base

Problem: Evaluate $\int 2^x\,dx$

Solution:

Using $\int a^x\,dx = \frac{a^x}{\ln a} + C$ :

\int 2^x\,dx = \frac{2^x}{\ln 2} + C

Example 5.27: Hyperbolic Function

Problem: Evaluate $\int \cosh x\,dx$

Solution:

Since $\frac{d}{dx}(\sinh x) = \cosh x$ :

\int \cosh x\,dx = \sinh x + C

Example 5.28: Composite Simplification

Problem: Evaluate $\int \frac{x^4 - 1}{x^2 + 1}\,dx$

Solution:

Factor: $x^4 - 1 = (x^2+1)(x^2-1)$

\frac{x^4 - 1}{x^2 + 1} = x^2 - 1

\int (x^2 - 1)\,dx = \frac{x^3}{3} - x + C

Example 5.29: Double Angle

Problem: Evaluate $\int \sin^2 x\,dx$

Solution:

Use identity $\sin^2 x = \frac{1 - \cos 2x}{2}$ :

\int \sin^2 x\,dx = \int \frac{1 - \cos 2x}{2}\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C

Example 5.30: Cosine Squared

Problem: Evaluate $\int \cos^2 x\,dx$

Solution:

Use identity $\cos^2 x = \frac{1 + \cos 2x}{2}$ :

\int \cos^2 x\,dx = \int \frac{1 + \cos 2x}{2}\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C

19. Verification Techniques

How to Verify Your Answers

Step 1: Differentiate your answer

If you found $\int f(x)\,dx = F(x) + C$ , compute $F'(x)$ .

Step 2: Compare with original

Check if $F'(x) = f(x)$ . If yes, your answer is correct.

Step 3: Don't forget the constant

The +C disappears when differentiating, so always include it in indefinite integrals.

Example 5.31: Verification Example

Claim: $\int x\cos x\,dx = x\sin x + \cos x + C$

Verification:

Differentiate the right side:

\frac{d}{dx}(x\sin x + \cos x) = \sin x + x\cos x - \sin x = x\cos x

This equals the integrand, so the answer is correct. ✓

20. Non-Elementary Integrals

Some functions have antiderivatives that cannot be expressed using elementary functions (polynomials, exponentials, logarithms, trigonometric functions, and their compositions). These are called non-elementary integrals.

Famous Examples:

$\int e^{-x^2}\,dx$ (Gaussian integral)

$\int \frac{\sin x}{x}\,dx$ (sine integral)

$\int \frac{e^x}{x}\,dx$ (exponential integral)

$\int \frac{1}{\ln x}\,dx$ (logarithmic integral)

$\int \sqrt{1+x^4}\,dx$ (elliptic integral)

$\int x^x\,dx$

These integrals exist (the functions have antiderivatives), but the antiderivatives cannot be written in closed form using elementary functions.

21. Final Practice Set

Test your understanding with these practice problems. Try to solve them without looking at the answers first.

Problem 1:

$\int (3x^5 - 2x^3 + x - 7)\,dx$

Answer

$\frac{x^6}{2} - \frac{x^4}{2} + \frac{x^2}{2} - 7x + C$

Problem 2:

$\int (2\sin x - 3\cos x)\,dx$

Answer

$-2\cos x - 3\sin x + C$

Problem 3:

$\int \frac{2}{x} + \frac{3}{x^2}\,dx$

Answer

$2\ln|x| - \frac{3}{x} + C$

Problem 4:

$\int (e^x + 5^x)\,dx$

Answer

$e^x + \frac{5^x}{\ln 5} + C$

Problem 5:

$\int \sec^2 x + \csc^2 x\,dx$

Answer

$\tan x - \cot x + C$

Problem 6:

$\int \frac{x^3 + 2x - 1}{x^2}\,dx$

Answer

$\frac{x^2}{2} + 2\ln|x| + \frac{1}{x} + C$

Problem 7 (IVP):

Find f(x) if $f'(x) = 6x^2 - 4x + 1$ and $f(1) = 5$ .

Answer

$f(x) = 2x^3 - 2x^2 + x + 4$

Problem 8:

$\int (\sqrt{x} - \sqrt[3]{x^2})\,dx$

Answer

$\frac{2}{3}x^{3/2} - \frac{3}{5}x^{5/3} + C$

22. Concept Summary

What is an Antiderivative?

A function F whose derivative equals f: $F'(x) = f(x)$

Why +C?

Antiderivatives are unique up to a constant. If F is one, F + C is also one.

When Does F Exist?

Continuous functions always have antiderivatives. Jump discontinuities prevent existence.

How to Find F?

Use basic formulas, algebraic manipulation, and later: substitution, by parts.

How to Verify?

Differentiate your answer. If you get back f(x), it's correct.

IVP Application

An initial condition F(a) = b picks out one specific antiderivative from F(x) + C.

Antiderivatives and Indefinite Integrals

1. Introduction

2. Definition of Antiderivative

3. Indefinite Integral Notation

4. Fundamental Properties

5. Existence of Antiderivatives

6. More Worked Examples

Frequently Asked Questions

What is the difference between antiderivative and indefinite integral?

Why do we add '+C' to indefinite integrals?

Does every function have an antiderivative?

What is the geometric meaning of an antiderivative?

Can antiderivatives be non-elementary?

What is a primitive function?

How do I verify F(x) is an antiderivative of f(x)?

Why is it called 'indefinite' integral?

What is the relationship between ∫ and d/dx?

Can discontinuous functions have antiderivatives?

7. Common Mistakes to Avoid

Mistake 1: Forgetting +C

Mistake 2: Wrong Power Rule

Mistake 3: Integrating 1/x

Mistake 4: Sign Error in Trig

Mistake 5: Multiple Constants

8. Study Tips

1. Think Backwards

2. Always Verify

3. Memorize Basics

4. Never Forget +C

5. Simplify First

6. Use Linearity

9. Quick Reference

Key Formulas

10. Additional Practice Examples

11. Challenge Problems

12. Geometric Interpretation

Slope Fields

Initial Value Problems

13. Historical Context

14. Summary Table

15. More Initial Value Problems

16. Extended Formula Table

17. Differentiation vs Integration

18. More Advanced Examples

19. Verification Techniques

How to Verify Your Answers

20. Non-Elementary Integrals

Famous Examples:

21. Final Practice Set

22. Concept Summary

What is an Antiderivative?

Why +C?

When Does F Exist?

How to Find F?

How to Verify?

IVP Application

Antiderivative

General Form

Uniqueness

Linearity

Remember