1. Introduction to Substitution

The substitution method (also called u-substitution or change of variables) is the integration counterpart to the chain rule for differentiation. If we know that:

\frac{d}{dx}[F(g(x))] = F'(g(x)) \cdot g'(x)

Then reversing this:

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

2. The Substitution Method

Theorem 5.13: Substitution Rule

If $u = g(x)$ is a differentiable function and $f$ is continuous, then:

\int f(g(x)) \cdot g'(x)\,dx = \int f(u)\,du

where we substitute back $u = g(x)$ after integration.

Remark 5.7: Step-by-Step Process

Choose u: Let u = g(x), an “inner function”
Find du: Compute du = g'(x) dx
Substitute: Replace all x expressions with u expressions
Integrate: Evaluate ∫f(u) du
Back-substitute: Replace u with g(x)

Example 5.36: Basic u-Substitution

Problem: Evaluate $\int 2x(x^2 + 1)^5\,dx$

Solution:

Let $u = x^2 + 1$ , then $du = 2x\,dx$

\int 2x(x^2 + 1)^5\,dx = \int u^5\,du = \frac{u^6}{6} + C = \frac{(x^2+1)^6}{6} + C

Example 5.37: Exponential Composite

Problem: Evaluate $\int e^{\sin x} \cos x\,dx$

Solution:

Let $u = \sin x$ , then $du = \cos x\,dx$

\int e^{\sin x} \cos x\,dx = \int e^u\,du = e^u + C = e^{\sin x} + C

Example 5.38: Adjusting Constants

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

Solution:

Let $u = x^2$ , then $du = 2x\,dx$ , so $x\,dx = \frac{1}{2}du$

\int x e^{x^2}\,dx = \frac{1}{2}\int e^u\,du = \frac{1}{2}e^u + C = \frac{1}{2}e^{x^2} + C

3. Common Substitution Patterns

Pattern 1: Power of a linear function

$\int (ax + b)^n\,dx$ → Let u = ax + b

= \frac{(ax+b)^{n+1}}{a(n+1)} + C

Pattern 2: Function times its derivative

$\int f(x)^n \cdot f'(x)\,dx$ → Let u = f(x)

= \frac{f(x)^{n+1}}{n+1} + C

Pattern 3: Exponential of a function

$\int e^{f(x)} \cdot f'(x)\,dx$ → Let u = f(x)

= e^{f(x)} + C

Pattern 4: Logarithmic form

$\int \frac{f'(x)}{f(x)}\,dx$ → Let u = f(x)

= \ln|f(x)| + C

Example 5.39: Logarithmic Pattern

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

Solution:

Recognize: numerator = derivative of denominator

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

Example 5.40: Tangent Function

Problem: Derive $\int \tan x\,dx$

Solution:

Write as $\frac{\sin x}{\cos x}$ . Let u = cos x, du = -sin x dx

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

4. Trigonometric Substitution

Theorem 5.14: Trigonometric Substitutions

Expression	Substitution	Identity Used
$\sqrt{a^2 - x^2}$	$x = a\sin\theta$	$1 - \sin^2\theta = \cos^2\theta$
$\sqrt{a^2 + x^2}$	$x = a\tan\theta$	$1 + \tan^2\theta = \sec^2\theta$
$\sqrt{x^2 - a^2}$	$x = a\sec\theta$	$\sec^2\theta - 1 = \tan^2\theta$

Example 5.41: Trig Substitution

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

Solution:

Let $x = 2\sin\theta$ , then $dx = 2\cos\theta\,d\theta$

$\sqrt{4-x^2} = \sqrt{4-4\sin^2\theta} = 2\cos\theta$

\int \frac{2\cos\theta}{2\cos\theta}\,d\theta = \int d\theta = \theta + C = \arcsin\frac{x}{2} + C

Example 5.42: Trig Substitution with √(a²+x²)

Problem: Evaluate $\int \frac{1}{\sqrt{x^2+9}}\,dx$

Solution:

Let $x = 3\tan\theta$ , then $dx = 3\sec^2\theta\,d\theta$

$\sqrt{x^2+9} = 3\sec\theta$

\int \frac{3\sec^2\theta}{3\sec\theta}\,d\theta = \int \sec\theta\,d\theta = \ln|\sec\theta + \tan\theta| + C

Back-substitute: $= \ln\left|\frac{x}{3} + \frac{\sqrt{x^2+9}}{3}\right| + C = \ln|x + \sqrt{x^2+9}| + C'$

Substitution Quiz

8

Questions

0

Correct

0%

Accuracy

1

\int 2x(x^2+1)^5\,dx

using

u = x^2+1

:

Easy

Not attempted

2

\int \cos x \cdot e^{\sin x}\,dx =

Easy

Not attempted

3

\int \frac{\ln x}{x}\,dx =

Medium

Not attempted

4

\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx =

Medium

Not attempted

5

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

Medium

Not attempted

6

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

Medium

Not attempted

7

For

\int \sqrt{a^2-x^2}\,dx

, what substitution?

Hard

Not attempted

8

\int \sec^3 x \tan x\,dx =

Medium

Not attempted

Frequently Asked Questions

How do I choose the right substitution?

Look for a function and its derivative together. Common patterns: f(g(x))·g'(x), composite functions, expressions under radicals.

What if du doesn't match exactly?

Often you can adjust by constants. If you need 2x dx but have x dx, multiply inside and divide outside by 2.

When should I use trig substitution?

For √(a²-x²) use x = a sin θ. For √(a²+x²) use x = a tan θ. For √(x²-a²) use x = a sec θ.

What's the most common mistake?

Forgetting to change dx to du, or not substituting back to x at the end.

Can I use substitution multiple times?

Yes! Some integrals require successive substitutions. Just keep track of all variables.

How do I know substitution is the right method?

If you see a composite function f(g(x)) and g'(x) (or a constant multiple) is present, try substitution.

5. Common Mistakes

Forgetting to change dx to du

Always replace dx with du using du = g'(x) dx.

Not substituting back

The final answer must be in terms of x, not u.

Wrong choice of u

Usually u should be the “inner” function whose derivative appears.

6. More Substitution Examples

Example 5.43: Exponential Composite

Problem: $\int x^2 e^{x^3}\,dx$

Let u = x³, du = 3x² dx, so x² dx = du/3:

\int x^2 e^{x^3}\,dx = \frac{1}{3}\int e^u\,du = \frac{1}{3}e^{x^3} + C

Example 5.44: Logarithm with Power

Problem: $\int \frac{(\ln x)^3}{x}\,dx$

Let u = ln x, du = dx/x:

\int (\ln x)^3 \cdot \frac{1}{x}\,dx = \int u^3\,du = \frac{u^4}{4} + C = \frac{(\ln x)^4}{4} + C

Example 5.45: Trigonometric Composite

Problem: $\int \sin^4 x \cos x\,dx$

Let u = sin x, du = cos x dx:

\int u^4\,du = \frac{u^5}{5} + C = \frac{\sin^5 x}{5} + C

Example 5.46: Rational with Sqrt

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

Let u = x²+1, du = 2x dx:

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

Example 5.47: Secant Times Tangent

Problem: $\int \sec^4 x \tan x\,dx$

Let u = sec x, du = sec x tan x dx:

\int \sec^3 x \cdot \sec x \tan x\,dx = \int u^3\,du = \frac{\sec^4 x}{4} + C

Example 5.48: Arctangent Composite

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

Let u = arctan x, du = dx/(1+x²):

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

7. Trigonometric Substitution Details

Example 5.49: √(a²-x²) Detailed

Problem: $\int \sqrt{9-x^2}\,dx$

Let x = 3 sin θ, dx = 3 cos θ dθ:

$\sqrt{9-x^2} = \sqrt{9-9\sin^2\theta} = 3\cos\theta$

\int 3\cos\theta \cdot 3\cos\theta\,d\theta = 9\int \cos^2\theta\,d\theta = \frac{9}{2}(\theta + \sin\theta\cos\theta) + C

Back-substitute: $\theta = \arcsin(x/3)$ , $\sin\theta = x/3$ , $\cos\theta = \sqrt{9-x^2}/3$

= \frac{9}{2}\arcsin\frac{x}{3} + \frac{x\sqrt{9-x^2}}{2} + C

Example 5.50: √(x²+a²) Detailed

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

Let x = 2 tan θ, dx = 2 sec²θ dθ:

$\sqrt{x^2+4} = 2\sec\theta$ , $x^2 = 4\tan^2\theta$

\int \frac{4\tan^2\theta \cdot 2\sec^2\theta}{2\sec\theta}\,d\theta = 4\int \tan^2\theta \sec\theta\,d\theta

Use tan²θ = sec²θ - 1:

= 4\int (\sec^3\theta - \sec\theta)\,d\theta

Example 5.51: √(x²-a²) Detailed

Problem: $\int \frac{\sqrt{x^2-9}}{x}\,dx$ (x > 3)

Let x = 3 sec θ, dx = 3 sec θ tan θ dθ:

$\sqrt{x^2-9} = 3\tan\theta$

\int \frac{3\tan\theta \cdot 3\sec\theta\tan\theta}{3\sec\theta}\,d\theta = 3\int \tan^2\theta\,d\theta = 3(\tan\theta - \theta) + C

Back: $= \sqrt{x^2-9} - 3\text{arcsec}\frac{x}{3} + C$

8. Practice Problems

1. $\int x(x^2+3)^{10}\,dx$

Answer

$\frac{(x^2+3)^{11}}{22} + C$

2. $\int e^{\sqrt{x}}/\sqrt{x}\,dx$

Answer

Let u = √x: $2e^{\sqrt{x}} + C$

3. $\int \frac{\cos(\ln x)}{x}\,dx$

Answer

$\sin(\ln x) + C$

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

Answer

Let u = eˣ: $\arctan(e^x) + C$

5. $\int \frac{x^3}{\sqrt{1-x^4}}\,dx$

Answer

Let u = x⁴: $-\frac{1}{2}\sqrt{1-x^4} + C$

6. $\int \tan^5 x \sec^2 x\,dx$

Answer

Let u = tan x: $\frac{\tan^6 x}{6} + C$

9. Advanced Examples

Example 5.52: Double Substitution

Problem: $\int \frac{1}{x\sqrt{\ln x}}\,dx$

Let u = ln x, then du = dx/x. Let v = √u, then dv = du/(2√u):

\int \frac{1}{\sqrt{u}}\,du = 2\sqrt{u} + C = 2\sqrt{\ln x} + C

Example 5.53: Completing the Square + Substitution

Problem: $\int \frac{1}{\sqrt{x^2+4x+8}}\,dx$

Complete: x²+4x+8 = (x+2)² + 4. Let u = x+2:

\int \frac{1}{\sqrt{u^2+4}}\,du = \ln|u + \sqrt{u^2+4}| + C = \ln|x+2 + \sqrt{x^2+4x+8}| + C

Example 5.54: Rationalizing Substitution

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

Let u = √x, so x = u², dx = 2u du:

\int \frac{2u}{1+u}\,du = \int \left(2 - \frac{2}{1+u}\right)\,du = 2u - 2\ln|1+u| + C

$= 2\sqrt{x} - 2\ln(1+\sqrt{x}) + C$

Example 5.55: Hyperbolic Substitution

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

Let x = sinh t, dx = cosh t dt:

$\sqrt{x^2+1} = \cosh t$

\int \cosh^2 t\,dt = \frac{t}{2} + \frac{\sinh t \cosh t}{2} + C = \frac{\sinh^{-1}x}{2} + \frac{x\sqrt{x^2+1}}{2} + C

10. Study Tips for Substitution

1. Look for f(g(x))·g'(x)

The derivative of the inner function should appear (perhaps with a constant).

2. Common u choices

Inner function, exponent, argument of trig/log, expression under radical.

3. Adjust constants

If du has extra constants, factor them out or multiply/divide as needed.

4. Trig sub for radicals

√(a²-x²) → x=a sin θ, √(a²+x²) → x=a tan θ, √(x²-a²) → x=a sec θ

5. Complete the square

For quadratics in denominator, complete square then substitute.

6. Always back-substitute

Final answer must be in terms of original variable.

11. Substitution Patterns Summary

Integrand Pattern	Substitution	Result Form
$f(ax+b)$	u = ax+b	$\frac{1}{a}F(ax+b)$
$f(g(x))g'(x)$	u = g(x)	$F(g(x))$
$\frac{g'(x)}{g(x)}$	u = g(x)	$\ln\|g(x)\|$
$\sqrt{a^2-x^2}$	x = a sin θ	Trig result
$\sqrt{a^2+x^2}$	x = a tan θ	Trig result
$\sqrt{x^2-a^2}$	x = a sec θ	Trig result

12. Additional Practice

Example 5.56: Power of ln

\int \frac{(\ln x)^5}{x}\,dx = \frac{(\ln x)^6}{6} + C

Example 5.57: Exponential Chain

\int x^3 e^{x^4}\,dx = \frac{e^{x^4}}{4} + C

Example 5.58: Trig Power

\int \cos^7 x \sin x\,dx = -\frac{\cos^8 x}{8} + C

Example 5.59: Inverse Trig

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

13. Challenge Problems

Challenge 1:

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

Solution

u = 1+x³: $-\frac{1}{3(1+x^3)} + C$

Challenge 2:

$\int e^x\sqrt{1+e^x}\,dx$

Solution

u = 1+eˣ: $\frac{2}{3}(1+e^x)^{3/2} + C$

Challenge 3:

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

Solution

u = cos x: $-\arctan(\cos x) + C$

Challenge 4:

$\int \frac{1}{x\sqrt{x^4-1}}\,dx$

Solution

u = x²: $\frac{1}{2}\text{arcsec}(x^2) + C$

14. More Worked Examples

Example 5.60: Exponential with Trig

Problem: $\int e^{\tan x} \sec^2 x\,dx$

Let u = tan x, du = sec²x dx:

= \int e^u\,du = e^{\tan x} + C

Example 5.61: Rational Function

Problem: $\int \frac{x^2}{1+x^6}\,dx$

Let u = x³, du = 3x² dx:

= \frac{1}{3}\int \frac{1}{1+u^2}\,du = \frac{1}{3}\arctan(x^3) + C

Example 5.62: Nested Functions

Problem: $\int \frac{\sin(\sqrt{x})}{\sqrt{x}}\,dx$

Let u = √x, du = dx/(2√x):

= 2\int \sin u\,du = -2\cos(\sqrt{x}) + C

Example 5.63: Product Form

Problem: $\int x(x^2-1)^{99}\,dx$

Let u = x²-1, du = 2x dx:

= \frac{1}{2}\int u^{99}\,du = \frac{(x^2-1)^{100}}{200} + C

Example 5.64: Inverse Trig Composite

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

Let u = arctan x, du = dx/(1+x²):

= \int e^u\,du = e^{\arctan x} + C

15. Final Practice Set

P1: $\int \frac{2x}{\sqrt{x^2+5}}\,dx$

Answer

$2\sqrt{x^2+5} + C$

P2: $\int \sin^6 x \cos x\,dx$

Answer

$\frac{\sin^7 x}{7} + C$

P3: $\int \frac{e^{2x}}{1+e^{4x}}\,dx$

Answer

$\frac{1}{2}\arctan(e^{2x}) + C$

P4: $\int \frac{(\ln x)^4}{x}\,dx$

Answer

$\frac{(\ln x)^5}{5} + C$

P5: $\int \sec^5 x \tan x\,dx$

Answer

$\frac{\sec^5 x}{5} + C$

P6: $\int x\sqrt{4-x^2}\,dx$

Answer

$-\frac{1}{3}(4-x^2)^{3/2} + C$

16. Common Substitution Summary

When You See	Try This u	Because du =
$f(x^n)\cdot x^{n-1}$	u = x^n	nx^{n-1}dx
$f(e^x)\cdot e^x$	u = e^x	e^x dx
$f(\ln x)/x$	u = ln x	dx/x
$f(\sin x)\cos x$	u = sin x	cos x dx
$f(\cos x)\sin x$	u = cos x	-sin x dx
$f(\tan x)\sec^2 x$	u = tan x	sec² x dx
$f(\sec x)\sec x \tan x$	u = sec x	sec x tan x dx
$f(\arctan x)/(1+x^2)$	u = arctan x	dx/(1+x²)

17. Additional Problems

\int \frac{x}{\sqrt{1-x^4}}\,dx = \frac{1}{2}\arcsin(x^2) + C

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

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

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

18. More Trig Substitution Examples

Example 5.65: ∫√(a²-x²) Type

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

Let x = 2 sin θ:

= 4\int \sin^2\theta\,d\theta = 2(\theta - \sin\theta\cos\theta) + C

$= 2\arcsin\frac{x}{2} - \frac{x\sqrt{4-x^2}}{2} + C$

Example 5.66: ∫1/√(x²+a²) Type

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

Let x = 3 tan θ:

= \frac{1}{9}\int \frac{\sec\theta}{\tan^2\theta}\,d\theta = -\frac{\sqrt{x^2+9}}{9x} + C

Example 5.67: ∫√(x²-a²) Type

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

Let x = 4 sec θ:

= 8\int \tan^2\theta\sec\theta\,d\theta

Result: $\frac{x\sqrt{x^2-16}}{2} - 8\ln|x + \sqrt{x^2-16}| + C$

19. Completing the Square with Substitution

Example 5.68: Quadratic in Denominator

Problem: $\int \frac{1}{\sqrt{x^2-6x+13}}\,dx$

Complete: x²-6x+13 = (x-3)² + 4

Let u = x-3:

= \int \frac{1}{\sqrt{u^2+4}}\,du = \ln|u + \sqrt{u^2+4}| + C

$= \ln|x-3 + \sqrt{x^2-6x+13}| + C$

Example 5.69: √(ax²+bx+c) Form

Problem: $\int \frac{1}{\sqrt{-x^2+4x+5}}\,dx$

Complete: -x²+4x+5 = 9-(x-2)²

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

20. Chapter Summary

Basic Substitution

For ∫f(g(x))g'(x)dx, let u = g(x). The integral becomes ∫f(u)du.

Trigonometric Substitution

For expressions with √(a²-x²), √(a²+x²), or √(x²-a²), use appropriate trig substitution.

Completing the Square

For quadratic expressions, complete the square first, then substitute.

Verification

Always differentiate your answer to verify it's correct.

21. Quick Reference Card

Power Functions

$f(x^n)x^{n-1} \to u=x^n$
$f(\sqrt{x})/\sqrt{x} \to u=\sqrt{x}$

Exponential/Log

$f(e^x)e^x \to u=e^x$
$f(\ln x)/x \to u=\ln x$

Trig

$f(\sin x)\cos x \to u=\sin x$
$f(\tan x)\sec^2 x \to u=\tan x$

22. More Practice Problems

P7: $\int \frac{x^3}{\sqrt{1+x^2}}\,dx$

Answer

$\frac{(1+x^2)^{3/2}}{3} - \sqrt{1+x^2} + C$

P8: $\int \frac{\arctan x}{1+x^2}\,dx$

Answer

$\frac{(\arctan x)^2}{2} + C$

P9: $\int \frac{\sin(\ln x)}{x}\,dx$

Answer

$-\cos(\ln x) + C$

P10: $\int x^2\sqrt{1-x^3}\,dx$

Answer

$-\frac{2(1-x^3)^{3/2}}{9} + C$

23. Challenge Problems

C5: $\int \frac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx$

Hint

Let u = x^(1/6), then x = u⁶

C6: $\int \sqrt{\frac{1+x}{1-x}}\,dx$

Hint

Try x = cos θ or rationalize

C7: $\int \frac{x}{\sqrt{x^4+x^2+1}}\,dx$

Hint

u = x² gives du = 2x dx

24. Detailed Worked Examples

Example 5.70: Rational with Square Root

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

Let u = √x, so x = u², dx = 2u du:

= 2\int \frac{u^2}{1+u^2}\,du = 2\int \left(1 - \frac{1}{1+u^2}\right)du

= 2u - 2\arctan u + C = 2\sqrt{x} - 2\arctan\sqrt{x} + C

Example 5.71: Multiple Substitutions

Problem: $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$

Let u = √x, du = dx/(2√x):

= 2\int e^u\,du = 2e^{\sqrt{x}} + C

25. Essential Substitution Results

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

\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C

\int f'(x)\cdot f(x)^n\,dx = \frac{f(x)^{n+1}}{n+1} + C

\int f'(x)e^{f(x)}\,dx = e^{f(x)} + C

26. Final Worked Examples

Example 5.72: Hyperbolic Substitution

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

Let x = sinh t, dx = cosh t dt:

= \int \frac{\cosh t}{\cosh t}\,dt = t + C = \text{arcsinh } x + C = \ln(x + \sqrt{x^2+1}) + C

Example 5.73: Euler Substitution

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

Let √(x²+1) - x = t, then x = (1-t²)/(2t):

= \sqrt{x^2+1} - x + C

27. Final Practice Problems

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

Hint

x = sin θ

$\int \frac{e^{2x}}{\sqrt{e^x+1}}\,dx$

Hint

u = eˣ+1

$\int \frac{\cos(\ln x)}{x}\,dx$

Answer

$\sin(\ln x) + C$

$\int \frac{x}{\sqrt{x^4-1}}\,dx$

Hint

u = x² gives du = 2x dx

28. Substitution Method Selection

Expression	Substitution
√(a²-x²)	x = a sin θ
√(a²+x²)	x = a tan θ or x = a sinh t
√(x²-a²)	x = a sec θ or x = a cosh t
f(g(x))·g'(x)	u = g(x)
√(ax²+bx+c)	Complete square first

29. Chapter Summary

Basic u-Sub

Identify inner function u and check if du is present

Trig Sub

For √(a²±x²), √(x²-a²) expressions

Complete Square

For ax²+bx+c before substitution

30. Important Standard Results

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

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

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

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

31. Final Examples

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

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

Integration by Substitution

1. Introduction to Substitution

2. The Substitution Method

3. Common Substitution Patterns

Pattern 1: Power of a linear function

Pattern 2: Function times its derivative

Pattern 3: Exponential of a function

Pattern 4: Logarithmic form

4. Trigonometric Substitution

Frequently Asked Questions

How do I choose the right substitution?

What if du doesn't match exactly?

When should I use trig substitution?

What's the most common mistake?

Can I use substitution multiple times?

How do I know substitution is the right method?

5. Common Mistakes

Forgetting to change dx to du

Not substituting back

Wrong choice of u

6. More Substitution Examples

7. Trigonometric Substitution Details

8. Practice Problems

9. Advanced Examples

10. Study Tips for Substitution

1. Look for f(g(x))·g'(x)

2. Common u choices

3. Adjust constants

4. Trig sub for radicals

5. Complete the square

6. Always back-substitute

11. Substitution Patterns Summary

12. Additional Practice

13. Challenge Problems

14. More Worked Examples

15. Final Practice Set

16. Common Substitution Summary

17. Additional Problems

18. More Trig Substitution Examples

19. Completing the Square with Substitution

20. Chapter Summary

Basic Substitution

Trigonometric Substitution

Completing the Square

Verification

21. Quick Reference Card

Power Functions

Exponential/Log

Trig

22. More Practice Problems

23. Challenge Problems

24. Detailed Worked Examples

25. Essential Substitution Results

26. Final Worked Examples

27. Final Practice Problems

28. Substitution Method Selection

29. Chapter Summary

Basic u-Sub

Trig Sub

Complete Square

30. Important Standard Results

31. Final Examples

Chain Rule in Reverse

Look for Patterns

Trig Substitution

Always Verify