Integration Techniques

Integration Techniques

Integration is the process of finding antiderivatives and calculating areas. Here's a comprehensive guide to the essential techniques:

Basic Integration Rules

Fundamental Rules

• $$\int k \, dx = kx + C$$ (where k is a constant)
• $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ (where n ≠ -1)
• $$\int \frac{1}{x} \, dx = \ln|x| + C$$
• $$\int e^x \, dx = e^x + C$$
• $$\int \sin x \, dx = -\cos x + C$$
• $$\int \cos x \, dx = \sin x + C$$
• $$\int \sec^2 x \, dx = \tan x + C$$
• $$\int \csc^2 x \, dx = -\cot x + C$$
• $$\int \sec x \tan x \, dx = \sec x + C$$
• $$\int \csc x \cot x \, dx = -\csc x + C$$

Linearity Properties

• $$\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$$
• $$\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$$ (where k is a constant)

Substitution Method (u-substitution)

Concept

Used when the integrand contains a function and its derivative.

If $$u = g(x)$$, then $$du = g'(x) \, dx$$, and:

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

Steps

1. Identify a substitution $$u = g(x)$$ such that $$du = g'(x) \, dx$$ appears in the integrand
2. Express the entire integrand in terms of u
3. Integrate with respect to u
4. Substitute back to express the answer in terms of x

Example

Evaluate $$\int x \cos(x^2) \, dx$$

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

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

Integration by Parts

Formula

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

Where $$u$$ and $$dv$$ are chosen from the integrand.

LIATE Rule for Choosing u

Choose u in this order of preference:

• Logarithmic functions: $$\ln x$$, $$\log_a x$$
• Inverse trigonometric functions: $$\arcsin x$$, $$\arctan x$$, etc.
• Algebraic functions: $$x^n$$, $$\sqrt{x}$$, etc.
• Trigonometric functions: $$\sin x$$, $$\cos x$$, etc.
• Exponential functions: $$e^x$$, $$a^x$$, etc.

Example

Evaluate $$\int x \ln x \, dx$$

Let $$u = \ln x$$ and $$dv = x \, dx$$

Then $$du = \frac{1}{x} \, dx$$ and $$v = \frac{x^2}{2}$$

$$\int x \ln x \, dx = \ln x \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \frac{x^2 \ln x}{2} - \int \frac{x}{2} \, dx$$

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

Trigonometric Integrals

Powers of Sine and Cosine

• $$\int \sin^n x \, dx$$: Use reduction formula or u-substitution
• $$\int \cos^n x \, dx$$: Use reduction formula or u-substitution
• For odd powers, separate one factor and use substitution
• For even powers, use double-angle formulas

Products of Sines and Cosines

• Use trigonometric identities to simplify
• $$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$
• $$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$$
• $$\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$$

Example

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

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

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

Trigonometric Substitution

When to Use

For integrals involving:

• $$\sqrt{a^2 - x^2}$$: Substitute $$x = a\sin \theta$$ (where $$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$$)
• $$\sqrt{a^2 + x^2}$$: Substitute $$x = a\tan \theta$$ (where $$-\frac{\pi}{2} < \theta < \frac{\pi}{2}$$)
• $$\sqrt{x^2 - a^2}$$: Substitute $$x = a\sec \theta$$ (where $$0 \leq \theta < \frac{\pi}{2}$$ or $$\pi \leq \theta < \frac{3\pi}{2}$$)

Example

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

Substitute $$x = 2\sin \theta$$, so $$dx = 2\cos \theta \, d\theta$$ and $$\sqrt{4-x^2} = \sqrt{4-4\sin^2 \theta} = 2\cos \theta$$

$$\int \frac{dx}{\sqrt{4-x^2}} = \int \frac{2\cos \theta \, d\theta}{2\cos \theta} = \int d\theta = \theta + C = \arcsin\left(\frac{x}{2}\right) + C$$

Partial Fractions

When to Use

For rational functions (ratio of polynomials) where the degree of the numerator is less than the degree of the denominator.

Steps

1. Factor the denominator completely
2. Set up partial fraction decomposition based on the factors:
   • For linear factors $$(ax+b)$$: $$\frac{A}{ax+b}$$
   • For repeated linear factors $$(ax+b)^n$$: $$\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + ... + \frac{A_n}{(ax+b)^n}$$
   • For irreducible quadratic factors $$(ax^2+bx+c)$$: $$\frac{Ax+B}{ax^2+bx+c}$$
   • For repeated quadratic factors $$(ax^2+bx+c)^n$$: $$\frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + ... + \frac{A_nx+B_n}{(ax^2+bx+c)^n}$$
3. Solve for the constants by:
   • Multiplying both sides by the denominator
   • Equating coefficients of like terms, or
   • Substituting specific values of x
4. Integrate each partial fraction term

Example

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

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

Set up partial fractions: $$\frac{5x+2}{(x-2)(x+1)} = \frac{A}{x-2} + \frac{B}{x+1}$$

Multiply by $$(x-2)(x+1)$$: $$5x+2 = A(x+1) + B(x-2)$$

Substitute $$x = 2$$: $$5(2)+2 = A(2+1) + B(0) \Rightarrow 12 = 3A \Rightarrow A = 4$$

Substitute $$x = -1$$: $$5(-1)+2 = A(0) + B(-1-2) \Rightarrow -3 = -3B \Rightarrow B = 1$$

$$\int \frac{5x+2}{x^2-x-2} \, dx = \int \left(\frac{4}{x-2} + \frac{1}{x+1}\right) \, dx = 4\ln|x-2| + \ln|x+1| + C = \ln|(x-2)^4(x+1)| + C$$

Integration of Special Functions

Integrals Involving $$\sqrt{a^2 \pm x^2}$$ and $$\sqrt{x^2 - a^2}$$

Use trigonometric substitution or consult a table of integrals.

Integrals Involving Rational Functions of Sine and Cosine

Substitute $$t = \tan(\frac{x}{2})$$, which gives:

• $$\sin x = \frac{2t}{1+t^2}$$
• $$\cos x = \frac{1-t^2}{1+t^2}$$
• $$dx = \frac{2}{1+t^2} \, dt$$

Common Pitfalls and Tips

• Always check your answer by differentiating it
• For u-substitution, make sure the du term appears in the integrand
• When using integration by parts, choose u and dv strategically
• For trigonometric integrals, use appropriate identities to simplify
• In partial fractions, ensure the numerator's degree is less than the denominator's (perform long division if necessary)
• Remember that different techniques can be combined for complex integrals
• Keep a table of common integrals handy for reference