Integration Techniques

Basic Integration

Understanding antiderivatives and fundamental integration techniques

Key Properties

Area above x-axis: positive contribution

Area below x-axis: negative contribution

\[\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\]

Reversing limits negates the integral

\[\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx\]

Additivity property

Basic Integration Rules

\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\]

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

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

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

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

Integration by Substitution

Using substitution to simplify complex integrals

The Process

Step 1

Identify u = g(x) where g'(x) appears in the integrand

Step 2

Express du = g'(x)dx

Step 3

Substitute and integrate in terms of u

Step 4

Substitute back to x

Example: U-Substitution

Evaluate \[\int x e^{x^2}\,dx\]

Step 1: Choose u

Let u = x²

Then du = 2x dx

Step 2: Rewrite

\[\frac{1}{2}\int e^u\,du\]

Step 3: Integrate

\[\frac{1}{2}e^u + C\]

Step 4: Substitute back

\[\frac{1}{2}e^{x^2} + C\]

Integration by Parts

A technique based on the product rule

The Formula

\[\int u\,dv = uv - \int v\,du\]

When to Use

  • Products where one factor is easier to differentiate
  • Common patterns: xeˣ, x ln(x), x sin(x)
  • Remember LIATE: Log, Inverse trig, Algebraic, Trig, Exponential

Example: Integration by Parts

Evaluate \[\int x\ln(x)\,dx\]

Step 1: Choose u and dv

u = ln(x)

dv = x dx

Step 2: Find du and v

du = \(\frac{1}{x}dx\)

v = \(\frac{x^2}{2}\)

Step 3: Apply Formula

\[\frac{x^2}{2}\ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x}dx\]

Step 4: Simplify

\[\frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C\]

Video Explanation

Practice Problems

Test your understanding of integration techniques.