1. Introduction

Just as we memorize differentiation formulas, we need a collection of basic integration formulas. These are the building blocks for evaluating indefinite integrals. Each formula comes from reversing a corresponding differentiation rule.

2. Power Rule for Integration

Theorem 5.6: Power Rule

For any real number $n \neq -1$ :

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

For $n = -1$ :

\int x^{-1}\,dx = \int \frac{1}{x}\,dx = \ln|x| + C

Proof:

Verify by differentiation:

\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = \frac{(n+1)x^n}{n+1} = x^n \quad \checkmark

∎

Example 5.32: Power Rule Applications

(a) $\int x^7\,dx = \frac{x^8}{8} + C$

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

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

(d) $\int \frac{1}{x^4}\,dx = -\frac{1}{3x^3} + C$

(e) $\int \frac{1}{\sqrt[3]{x}}\,dx = \frac{3}{2}\sqrt[3]{x^2} + C$

3. Exponential and Logarithmic Functions

Theorem 5.7: Exponential Integrals

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

\int a^x\,dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)

\int e^{ax}\,dx = \frac{e^{ax}}{a} + C \quad (a \neq 0)

\int \frac{1}{x}\,dx = \ln|x| + C

Example 5.33: Exponential Integrals

(a) $\int e^{5x}\,dx = \frac{e^{5x}}{5} + C$

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

(c) $\int e^{-x}\,dx = -e^{-x} + C$

(d) $\int 10^x\,dx = \frac{10^x}{\ln 10} + C$

4. Trigonometric Functions

Theorem 5.8: Basic Trigonometric Integrals

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

\int \cos x\,dx = \sin x + C

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

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

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

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

Theorem 5.9: tan, cot, sec, csc Integrals

\int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C

\int \cot x\,dx = \ln|\sin x| + C

\int \sec x\,dx = \ln|\sec x + \tan x| + C

\int \csc x\,dx = -\ln|\csc x + \cot x| + C

Example 5.34: Trig Integrals

(a) $\int 3\sin x\,dx = -3\cos x + C$

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

(c) $\int \tan 3x\,dx = \frac{1}{3}\ln|\sec 3x| + C$

5. Inverse Trigonometric Functions

Theorem 5.10: Inverse Trig Integrals

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

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

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

Theorem 5.11: Generalized Forms

For $a > 0$ :

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

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

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

Example 5.35: Inverse Trig Integrals

(a) $\int \frac{1}{\sqrt{9-x^2}}\,dx = \arcsin\frac{x}{3} + C$

(b) $\int \frac{1}{4+x^2}\,dx = \frac{1}{2}\arctan\frac{x}{2} + C$

(c) $\int \frac{1}{x^2-9}\,dx = \frac{1}{6}\ln\left|\frac{x-3}{x+3}\right| + C$

6. Hyperbolic Functions

Definition 5.3: Hyperbolic Functions

\sinh x = \frac{e^x - e^{-x}}{2}

\cosh x = \frac{e^x + e^{-x}}{2}

Theorem 5.12: Hyperbolic Integrals

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

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

\int \text{sech}^2 x\,dx = \tanh x + C

\int \frac{1}{\sqrt{x^2+1}}\,dx = \ln|x + \sqrt{x^2+1}| + C

\int \frac{1}{\sqrt{x^2-1}}\,dx = \ln|x + \sqrt{x^2-1}| + C \quad (x > 1)

7. Complete Formula Reference

Power & Logarithmic

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

\int \frac{1}{x}\,dx = \ln|x| + C

Exponential

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

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

Trigonometric

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

\int \cos x\,dx = \sin x + C

\int \tan x\,dx = \ln|\sec x| + C

\int \cot x\,dx = \ln|\sin x| + C

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

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

Inverse Trigonometric

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

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

Basic Formulas Quiz

8

Questions

0

Correct

0%

Accuracy

1

\int x^5\,dx =

Easy

Not attempted

2

\int e^{3x}\,dx =

Easy

Not attempted

3

\int \tan x\,dx =

Medium

Not attempted

4

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

Easy

Not attempted

5

\int 3^x\,dx =

Medium

Not attempted

6

\int \cosh x\,dx =

Medium

Not attempted

7

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

Medium

Not attempted

8

\int \sec x\,dx =

Hard

Not attempted

Frequently Asked Questions

How do I know which formula to use?

Identify the form of the integrand. Look for: power functions (x^n), exponentials (e^x, a^x), trig functions, inverse trig patterns (1/√(1-x²), 1/(1+x²)), and logarithms (1/x).

Why is ∫1/x dx = ln|x| and not ln(x)?

The absolute value is needed because ln(x) is only defined for x > 0, but 1/x is defined for all x ≠ 0.

What's the difference between ∫sin x dx and ∫sin(ax) dx?

∫sin x dx = -cos x + C, but ∫sin(ax) dx = -cos(ax)/a + C. The chain rule in reverse requires dividing by a.

How do I integrate tan x and cot x?

What are the integrals of sec x and csc x?

∫sec x dx = ln|sec x + tan x| + C. ∫csc x dx = -ln|csc x + cot x| + C.

When do I use arcsin vs arccos?

∫1/√(1-x²) dx = arcsin x + C. Usually arcsin is preferred.

What about integrals with √(a²-x²)?

∫1/√(a²-x²) dx = arcsin(x/a) + C. These generalize the basic formulas.

How do hyperbolic functions relate to exponentials?

sinh x = (e^x - e^{-x})/2 and cosh x = (e^x + e^{-x})/2.

8. Common Mistakes

Wrong sign for sin/cos

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

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

Missing coefficient for e^{ax}

Wrong: $\int e^{2x}\,dx = e^{2x} + C$

Right: $\int e^{2x}\,dx = \frac{e^{2x}}{2} + C$

Missing ln in a^x

Wrong: $\int 2^x\,dx = 2^x + C$

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

Missing 1/a in generalized formulas

Wrong: $\int \frac{1}{4+x^2}\,dx = \arctan\frac{x}{2} + C$

Right: $\int \frac{1}{4+x^2}\,dx = \frac{1}{2}\arctan\frac{x}{2} + C$

9. More Worked Examples

Example 5.36: Polynomial

Problem: $\int (3x^4 - 2x^3 + 5x - 7)\,dx$

= \frac{3x^5}{5} - \frac{x^4}{2} + \frac{5x^2}{2} - 7x + C

Example 5.37: Fractional Powers

Problem: $\int (\sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x})\,dx$

Write as: $x^{1/2} + x^{1/3} + x^{1/4}$

= \frac{2x^{3/2}}{3} + \frac{3x^{4/3}}{4} + \frac{4x^{5/4}}{5} + C

Example 5.38: Negative Powers

Problem: $\int \frac{1}{x^5}\,dx$

\int x^{-5}\,dx = \frac{x^{-4}}{-4} + C = -\frac{1}{4x^4} + C

Example 5.39: Exponential Sum

Problem: $\int (e^x + 2^x + 3^x)\,dx$

= e^x + \frac{2^x}{\ln 2} + \frac{3^x}{\ln 3} + C

Example 5.40: Trig Sum

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

= -3\cos x - 2\sin x + \tan x + C

Example 5.41: Rational Simplification

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

Simplify: $= x + 2 - x^{-1} + x^{-2}$

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

Example 5.42: Inverse Trig Pattern

Problem: $\int \frac{3}{\sqrt{25-x^2}}\,dx$

Here a = 5:

= 3\arcsin\frac{x}{5} + C

Example 5.43: Arctan Form

Problem: $\int \frac{2}{9+x^2}\,dx$

Here a = 3:

= \frac{2}{3}\arctan\frac{x}{3} + C

10. Study Tips

1. Memorize Core Formulas

Power, exponential, basic trig, and inverse trig forms.

2. Verify by Differentiating

Always check your answer by taking the derivative.

3. Watch Linear Arguments

∫sin(ax)dx adds a factor of 1/a.

4. Recognize Patterns

1/(1+x²) → arctan, 1/√(1-x²) → arcsin

5. Simplify First

Algebra often reveals a known form.

6. Use Flash Cards

Practice until formulas are automatic.

11. Extended Practice

Example 5.44: Mixed Problem 1

Problem: $\int \left(e^{2x} + \frac{1}{2x} + \cos 3x\right)\,dx$

= \frac{e^{2x}}{2} + \frac{1}{2}\ln|x| + \frac{\sin 3x}{3} + C

Example 5.45: Mixed Problem 2

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

Simplify: $= x^2 - x^{-2}$

= \frac{x^3}{3} + \frac{1}{x} + C

Example 5.46: Mixed Problem 3

Problem: $\int (\sec x \tan x + \csc x \cot x)\,dx$

= \sec x - \csc x + C

Example 5.47: Mixed Problem 4

Problem: $\int 5^{2x}\,dx$

Let u = 2x, so dx = du/2:

= \frac{1}{2} \cdot \frac{5^{2x}}{\ln 5} + C = \frac{5^{2x}}{2\ln 5} + C

Example 5.48: Mixed Problem 5

Problem: $\int \frac{4}{\sqrt{16-x^2}}\,dx$

= 4\arcsin\frac{x}{4} + C

Example 5.49: Mixed Problem 6

Problem: $\int \frac{1}{x^2 - 25}\,dx$

Here a = 5:

= \frac{1}{10}\ln\left|\frac{x-5}{x+5}\right| + C

Example 5.50: Hyperbolic

Problem: $\int (2\sinh x + 3\cosh x)\,dx$

= 2\cosh x + 3\sinh x + C

Example 5.51: Complex Fraction

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

Simplify: $= 1 + x^{-2}$

= x - \frac{1}{x} + C

12. Complete Formula Summary

f(x)	∫f(x)dx	Notes
$x^n$	$\frac{x^{n+1}}{n+1}+C$	n ≠ -1
$1/x$	$\ln\|x\|+C$	x ≠ 0
$e^x$	$e^x+C$	-
$a^x$	$a^x/\ln a+C$	a > 0, a ≠ 1
$\sin x$	$-\cos x+C$	-
$\cos x$	$\sin x+C$	-
$\sec^2 x$	$\tan x+C$	-
$\csc^2 x$	$-\cot x+C$	-
$\tan x$	$\ln\|\sec x\|+C$	-
$\cot x$	$\ln\|\sin x\|+C$	-
$\sec x$	$\ln\|\sec x+\tan x\|+C$	-
$\csc x$	$-\ln\|\csc x+\cot x\|+C$	-
$1/\sqrt{1-x^2}$	$\arcsin x+C$	\|x\| < 1
$1/(1+x^2)$	$\arctan x+C$	-
$1/\sqrt{a^2-x^2}$	$\arcsin(x/a)+C$	\|x\| < a
$1/(a^2+x^2)$	$(1/a)\arctan(x/a)+C$	-
$\sinh x$	$\cosh x+C$	-
$\cosh x$	$\sinh x+C$	-

13. Self-Test Problems

Problem 1:

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

Answer

$\frac{x^7}{7} - \frac{3x^5}{5} + \frac{2x^3}{3} - 5x + C$

Problem 2:

$\int (4e^{-x} + 3 \cdot 5^x)\,dx$

Answer

$-4e^{-x} + \frac{3 \cdot 5^x}{\ln 5} + C$

Problem 3:

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

Answer

$-\frac{\cos 2x}{2} + \frac{\sin 3x}{3} + C$

Problem 4:

$\int \frac{5}{\sqrt{49-x^2}}\,dx$

Answer

$5\arcsin\frac{x}{7} + C$

Problem 5:

$\int \frac{3}{16+x^2}\,dx$

Answer

$\frac{3}{4}\arctan\frac{x}{4} + C$

Problem 6:

$\int (\tan x + \cot x)\,dx$

Answer

$\ln|\sec x| + \ln|\sin x| + C$

14. Deriving the Formulas

Proof:

Derivation of ∫sec x dx

Multiply by (sec x + tan x)/(sec x + tan x):

\int \sec x\,dx = \int \frac{\sec x(\sec x + \tan x)}{\sec x + \tan x}\,dx = \int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x}\,dx

Let u = sec x + tan x, then du = (sec x tan x + sec²x) dx:

= \int \frac{du}{u} = \ln|u| + C = \ln|\sec x + \tan x| + C \quad \checkmark

∎

Proof:

Derivation of ∫cot x dx

Write cot x = cos x / sin x. Let u = sin x, du = cos x dx:

\int \frac{\cos x}{\sin x}\,dx = \int \frac{du}{u} = \ln|u| + C = \ln|\sin x| + C \quad \checkmark

∎

Proof:

Why ∫1/(a²+x²) dx = (1/a)arctan(x/a)

Let x = a tan θ, then dx = a sec²θ dθ:

\int \frac{1}{a^2 + a^2\tan^2\theta} \cdot a\sec^2\theta\,d\theta = \int \frac{a\sec^2\theta}{a^2\sec^2\theta}\,d\theta = \frac{1}{a}\int d\theta = \frac{\theta}{a} + C

Since θ = arctan(x/a): $= \frac{1}{a}\arctan\frac{x}{a} + C \quad \checkmark$

∎

15. Historical Notes

The integration formulas we use today were developed over centuries by mathematicians including Newton, Leibniz, Euler, and many others.

Leonhard Euler (1707-1783) made enormous contributions to integration, including the development of hyperbolic functions and their integrals.

The notation ∫ was introduced by Leibniz in 1675, representing an elongated “S” for “summa” (sum), reflecting the connection between integration and summation.

16. Linear Arguments

Remark 5.9: Linear Argument Rule

For $\int f(ax+b)\,dx$ , if $\int f(x)\,dx = F(x) + C$ , then:

\int f(ax+b)\,dx = \frac{1}{a}F(ax+b) + C

Example 5.52: Sin with Linear Argument

Problem: $\int \sin(3x+2)\,dx$

= -\frac{1}{3}\cos(3x+2) + C

Example 5.53: Exponential with Linear Argument

Problem: $\int e^{4x-1}\,dx$

= \frac{1}{4}e^{4x-1} + C

Example 5.54: Power with Linear Argument

Problem: $\int (2x+5)^7\,dx$

= \frac{1}{2} \cdot \frac{(2x+5)^8}{8} + C = \frac{(2x+5)^8}{16} + C

Example 5.55: Sec² with Linear Argument

Problem: $\int \sec^2(5x)\,dx$

= \frac{1}{5}\tan(5x) + C

Example 5.56: Reciprocal with Linear Argument

Problem: $\int \frac{1}{3x+7}\,dx$

= \frac{1}{3}\ln|3x+7| + C

Example 5.57: Arcsin Form with Linear

Problem: $\int \frac{1}{\sqrt{1-(2x)^2}}\,dx = \int \frac{1}{\sqrt{1-4x^2}}\,dx$

= \frac{1}{2}\arcsin(2x) + C

17. Challenge Problems

Challenge 1:

$\int \frac{x^5 + x^3 - x + 2}{x^3}\,dx$

Solution

Simplify: $x^2 + 1 - x^{-2} + 2x^{-3}$

$= \frac{x^3}{3} + x + \frac{1}{x} - \frac{1}{x^2} + C$

Challenge 2:

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

Solution

Expand: $e^{2x} - 2 + e^{-2x}$

$= \frac{e^{2x}}{2} - 2x - \frac{e^{-2x}}{2} + C$

Challenge 3:

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

Solution

Simplify using sin²x + cos²x = 1:

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

Challenge 4:

$\int \frac{1+\tan^2 x}{\sec^2 x}\,dx$

Solution

Since 1 + tan²x = sec²x:

$= \int 1\,dx = x + C$

Challenge 5:

$\int \frac{3}{\sqrt{9-4x^2}}\,dx$

Solution

Write as $\frac{3}{\sqrt{9(1-\frac{4x^2}{9})}} = \frac{1}{\sqrt{1-(\frac{2x}{3})^2}}$

$= \frac{3}{2}\arcsin\frac{2x}{3} + C$

18. Applications

Example 5.58: Position from Velocity

Problem: A particle has velocity v(t) = 3t² - 2t + 1. Find position s(t) if s(0) = 5.

Solution:

s(t) = \int (3t^2 - 2t + 1)\,dt = t^3 - t^2 + t + C

Using s(0) = 5: C = 5, so $s(t) = t^3 - t^2 + t + 5$

Example 5.59: Marginal Cost

Problem: Marginal cost is C'(x) = 100 + 0.5x. Find total cost if fixed costs are $2000.

Solution:

C(x) = \int (100 + 0.5x)\,dx = 100x + 0.25x^2 + C

Using C(0) = 2000: $C(x) = 100x + 0.25x^2 + 2000$

Example 5.60: Decay Rate

Problem: The rate of decay is dN/dt = -0.05N. Given N(0) = 1000, this is solved by:

N(t) = 1000e^{-0.05t}

The integral $\int e^{-0.05t}\,dt = -20e^{-0.05t} + C$ is used in the solution process.

19. Additional Practice

Example 5.61: Mixed 1

\int (x^{-1/2} + x^{1/2} + x^{3/2})\,dx = 2\sqrt{x} + \frac{2x\sqrt{x}}{3} + \frac{2x^2\sqrt{x}}{5} + C

Example 5.62: Mixed 2

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

Example 5.63: Mixed 3

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

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

Example 5.64: Mixed 4

\int \frac{1}{\sqrt{x}(1+x)}\,dx

Let u = √x, then x = u², dx = 2u du:

= \int \frac{2u}{u(1+u^2)}\,du = 2\arctan(\sqrt{x}) + C

20. Quick Reference Card

Power

$\int x^n = \frac{x^{n+1}}{n+1}$

$\int \frac{1}{x} = \ln|x|$

Exponential

$\int e^x = e^x$

$\int a^x = \frac{a^x}{\ln a}$

Trig Basic

$\int \sin x = -\cos x$

$\int \cos x = \sin x$

Trig Squares

$\int \sec^2 x = \tan x$

$\int \csc^2 x = -\cot x$

Inverse Trig

$\int \frac{1}{\sqrt{1-x^2}} = \arcsin x$

$\int \frac{1}{1+x^2} = \arctan x$

Hyperbolic

$\int \sinh x = \cosh x$

$\int \cosh x = \sinh x$

21. Final Practice Set

1. $\int (7x^6 - 4x^3 + 2)\,dx$

Answer

$x^7 - x^4 + 2x + C$

2. $\int (5e^{3x} - 2 \cdot 4^x)\,dx$

Answer

$\frac{5e^{3x}}{3} - \frac{2 \cdot 4^x}{\ln 4} + C$

3. $\int (\sin 4x - \cos 2x)\,dx$

Answer

$-\frac{\cos 4x}{4} - \frac{\sin 2x}{2} + C$

4. $\int \frac{6}{\sqrt{36-x^2}}\,dx$

Answer

$6\arcsin\frac{x}{6} + C$

5. $\int \frac{4}{25+x^2}\,dx$

Answer

$\frac{4}{5}\arctan\frac{x}{5} + C$

6. $\int (3\sec^2 x + 2\csc^2 x)\,dx$

Answer

$3\tan x - 2\cot x + C$

7. $\int \frac{x^4+x^2-1}{x^2}\,dx$

Answer

$\frac{x^3}{3} + x + \frac{1}{x} + C$

8. $\int (5\sinh x - 3\cosh x)\,dx$

Answer

$5\cosh x - 3\sinh x + C$

22. Differentiation vs Integration

Function	Derivative	Integral
$x^n$	$nx^{n-1}$	$\frac{x^{n+1}}{n+1}+C$
$e^x$	$e^x$	$e^x+C$
$\ln x$	$1/x$	$x\ln x - x + C$
$\sin x$	$\cos x$	$-\cos x+C$
$\cos x$	$-\sin x$	$\sin x+C$
$\tan x$	$\sec^2 x$	$\ln\|\sec x\|+C$
$\arctan x$	$\frac{1}{1+x^2}$	$x\arctan x - \frac{1}{2}\ln(1+x^2)+C$

23. More Challenge Problems

Example 5.65: Algebraic Manipulation

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

Expand: $= \int \frac{x^2+2x+1}{x}\,dx = \int (x + 2 + \frac{1}{x})\,dx$

= \frac{x^2}{2} + 2x + \ln|x| + C

Example 5.66: Trig Identity

Problem: $\int \frac{1}{\sin^2 x \cos^2 x}\,dx$

Use: $\sin^2 x \cos^2 x = \frac{\sin^2 2x}{4}$

= 4\int \csc^2 2x\,dx = -2\cot 2x + C

Example 5.67: Completing Square

Problem: $\int \frac{1}{x^2 - 4x + 13}\,dx$

Complete: $x^2 - 4x + 13 = (x-2)^2 + 9$

= \frac{1}{3}\arctan\frac{x-2}{3} + C

Example 5.68: Product Expansion

Problem: $\int (e^x + 1)(e^x - 1)\,dx = \int (e^{2x} - 1)\,dx$

= \frac{e^{2x}}{2} - x + C

Example 5.69: Hyperbolic Identity

Problem: $\int \tanh x\,dx = \int \frac{\sinh x}{\cosh x}\,dx$

Let u = cosh x, du = sinh x dx:

= \ln|\cosh x| + C = \ln(\cosh x) + C

Example 5.70: Sum of Squares

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

Write: $\frac{x^2}{1+x^2} = 1 - \frac{1}{1+x^2}$

= x - \arctan x + C

24. Even More Practice

Problem A:

$\int (x^{3/2} - x^{-3/2})\,dx$

$= \frac{2x^{5/2}}{5} + 2x^{-1/2} + C$

Problem B:

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

$= \frac{e^x - e^{-x}}{2} + C = \sinh x + C$

Problem C:

$\int (1 + \tan^2 x)\,dx$

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

Problem D:

$\int \frac{2}{\sqrt{100-x^2}}\,dx$

$= 2\arcsin\frac{x}{10} + C$

Problem E:

$\int \frac{5}{49+x^2}\,dx$

$= \frac{5}{7}\arctan\frac{x}{7} + C$

Problem F:

$\int (7^x + \ln 7)\,dx$

$= \frac{7^x}{\ln 7} + x\ln 7 + C$

25. Concept Map

Algebraic

• Power rule: $x^n \to \frac{x^{n+1}}{n+1}$
• Reciprocal: $\frac{1}{x} \to \ln|x|$
• Simplify first when possible

Exponential

• Natural: $e^x \to e^x$
• General: $a^x \to \frac{a^x}{\ln a}$
• Linear arg: divide by coefficient

Trigonometric

• sin/cos: sign changes
• tan/cot: log results
• sec/csc: complex log forms

Inverse Trig

• $\frac{1}{\sqrt{1-x^2}} \to \arcsin$
• $\frac{1}{1+x^2} \to \arctan$
• Generalized forms with a

Hyperbolic

• $\sinh x \to \cosh x$
• $\cosh x \to \sinh x$
• Related to exponentials

Tips

• Always add + C
• Verify by differentiating
• Simplify before integrating

Final Quick Reference

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

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

Basic Integration Formulas

1. Introduction

2. Power Rule for Integration

3. Exponential and Logarithmic Functions

4. Trigonometric Functions

5. Inverse Trigonometric Functions

6. Hyperbolic Functions

7. Complete Formula Reference

Power & Logarithmic

Exponential

Trigonometric

Inverse Trigonometric

Frequently Asked Questions

How do I know which formula to use?

Why is ∫1/x dx = ln|x| and not ln(x)?

What's the difference between ∫sin x dx and ∫sin(ax) dx?

How do I integrate tan x and cot x?

What are the integrals of sec x and csc x?

When do I use arcsin vs arccos?

What about integrals with √(a²-x²)?

How do hyperbolic functions relate to exponentials?

8. Common Mistakes

Wrong sign for sin/cos

Missing coefficient for e^{ax}

Missing ln in a^x

Missing 1/a in generalized formulas

9. More Worked Examples

10. Study Tips

1. Memorize Core Formulas

2. Verify by Differentiating

3. Watch Linear Arguments

4. Recognize Patterns

5. Simplify First

6. Use Flash Cards

11. Extended Practice

12. Complete Formula Summary

13. Self-Test Problems

14. Deriving the Formulas

Derivation of ∫sec x dx

Derivation of ∫cot x dx

Why ∫1/(a²+x²) dx = (1/a)arctan(x/a)

15. Historical Notes

16. Linear Arguments

17. Challenge Problems

18. Applications

19. Additional Practice

20. Quick Reference Card

Power

Exponential

Trig Basic

Trig Squares

Inverse Trig

Hyperbolic

21. Final Practice Set

22. Differentiation vs Integration

23. More Challenge Problems

24. Even More Practice

25. Concept Map

Algebraic

Exponential

Trigonometric

Inverse Trig

Hyperbolic

Tips

Power Rule

Exponential

Inverse Trig

Verification

Final Quick Reference