Comprehensive Integration Techniques Guide

Comprehensive Integration Techniques

Master all essential integration methods with detailed explanations and examples

Integration Fundamentals

Integration is the reverse process of differentiation. The indefinite integral of a function \(f(x)\) is:

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

where \(F'(x) = f(x)\) and \(C\) is the constant of integration.

1. Basic Integration Formulas

Power Rule

\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Example:

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

Exponential Functions

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

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

Trigonometric Functions

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

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

2. Substitution Rule (U-Substitution)

The Substitution Method

Used when the integrand is a composite function \(f(g(x))\) multiplied by its derivative \(g'(x)\):

\displaystyle\int f(g(x))g'(x)\,dx = \int f(u)\,du \quad \text{where } u = g(x), du = g'(x)dx

Example 1:

\displaystyle\int 2x\cos(x^2)\,dx

Let \(u = x^2\), then \(du = 2x\,dx\):

= \displaystyle\int \cos u\,du = \sin u + C = \sin(x^2) + C

Example 2:

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

Let \(u = \ln x\), then \(du = \frac{1}{x}dx\):

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

3. Integration by Parts

The Formula

\displaystyle\int u\,dv = uv - \int v\,du

The LIATE rule helps choose \(u\) (the part to differentiate):

Logarithmic (\(\ln x\))
Inverse Trigonometric (\(\arctan x\))
Algebraic (\(x^n\))
Trigonometric (\(\sin x\))
Exponential (\(e^x\))

Example 1:

\displaystyle\int x e^x\,dx

Let \(u = x\) (algebraic), \(dv = e^x\,dx\):

du = dx\), \(v = e^x

= x e^x - \displaystyle\int e^x\,dx = x e^x - e^x + C = e^x(x - 1) + C

Example 2:

\displaystyle\int x^2 \sin x\,dx

Requires integration by parts twice.

4. Trigonometric Integrals

Powers of Sine and Cosine

Odd Power of Sine

\displaystyle\int \sin^{2k+1}x \cos^n x\,dx

Save one \(\sin x\) and convert rest to \(\cos x\) using \(\sin^2x = 1 - \cos^2x\)

Odd Power of Cosine

\displaystyle\int \sin^n x \cos^{2k+1}x\,dx

Save one \(\cos x\) and convert rest to \(\sin x\) using \(\cos^2x = 1 - \sin^2x\)

Example:

\displaystyle\int \sin^3x \cos^2x\,dx

Convert \(\sin^3x = \sin x(1 - \cos^2x)\):

= \displaystyle\int (1 - \cos^2x)\cos^2x \sin x\,dx

Let \(u = \cos x\), \(du = -\sin x\,dx\):

= -\displaystyle\int (1 - u^2)u^2\,du = \frac{u^5}{5} - \frac{u^3}{3} + C = \frac{\cos^5x}{5} - \frac{\cos^3x}{3} + C

Powers of Secant and Tangent

Even Power of Secant

\displaystyle\int \sec^{2k}x \tan^n x\,dx

Save \(\sec^2x\) and convert rest to \(\tan x\) using \(\sec^2x = 1 + \tan^2x\)

Odd Power of Tangent

\(\displaystyle\int \sec^n x \tan^{2k+1}x\,dx\"

Save \(\sec x \tan x\) and convert rest to \(\sec x\) using \(\tan^2x = \sec^2x - 1\)

5. Trigonometric Substitution

Three Standard Cases

Integral Contains	Substitution	Identity Used
\(\sqrt{a^2 - x^2}\)	\(x = a\sin\theta\), \(dx = a\cos\theta\,d\theta\)	\(1 - \sin^2\theta = \cos^2\theta\)
\(\sqrt{a^2 + x^2}\)	\(x = a\tan\theta\), \(dx = a\sec^2\theta\,d\theta\)	\(1 + \tan^2\theta = \sec^2\theta\)
\(\sqrt{x^2 - a^2}\)	\(x = a\sec\theta\), \(dx = a\sec\theta\tan\theta\,d\theta\)	\(\sec^2\theta - 1 = \tan^2\theta\)

Example:

\displaystyle\int \frac{dx}{\sqrt{x^2 + 4}}

Let \(x = 2\tan\theta\), \(dx = 2\sec^2\theta\,d\theta\):

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

Convert back using \(\tan\theta = \frac{x}{2}\):

= \ln\left|\frac{\sqrt{x^2 + 4}}{2} + \frac{x}{2}\right| + C

6. Partial Fractions

Rational Function Decomposition

Used for integrals of rational functions \(\frac{P(x)}{Q(x)}\) where \(\deg(P) < \deg(Q)\):

\displaystyle\int \frac{P(x)}{Q(x)}\,dx

Case 1: Distinct Linear Factors

\displaystyle\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}

Case 2: Repeated Linear Factors

\displaystyle\frac{1}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}

Example:

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

Decompose:

\displaystyle\frac{2x + 3}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}

Solve for A and B:

\(A = -1, B = 3\)

= -\ln|x+1| + 3\ln|x+2| + C

7. Special Integration Techniques

Weierstrass Substitution

For integrals with rational trigonometric functions:

t = \tan\left(\frac{x}{2}\right)

\sin x = \frac{2t}{1 + t^2}, \quad \cos x = \frac{1 - t^2}{1 + t^2}

dx = \frac{2}{1 + t^2}dt

Improper Integrals

\displaystyle\int_a^\infty f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx

\displaystyle\int_{-\infty}^b f(x)\,dx = \lim_{a\to-\infty}\int_a^b f(x)\,dx

\displaystyle\int_{-\infty}^\infty f(x)\,dx = \int_{-\infty}^c f(x)\,dx + \int_c^\infty f(x)\,dx

Practice Problems

Identify the Technique

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

\displaystyle\int x^2 e^{x^3}\,dx

\displaystyle\int \frac{1}{x^2 + 9}\,dx

Integration Quiz

Which method for \(\displaystyle\int x \sin x\,dx\)?

Substitution Integration by Parts Partial Fractions

The integral \(\displaystyle\int \frac{1}{x^2 - 4}\,dx\) requires:

Trig Substitution Partial Fractions Integration by Parts