Techniques for Solving Integrals

There are several powerful strategies available for evaluating integrals, each suited to a particular form of the integrand.

Method 1
Substitution Method
When a substitution $ x = g(t) $ is introduced, the integral can be rewritten as: $$ \int f(x) \: dx = \int f(g(t)) \cdot g'(t) \: dt $$ This change of variables can simplify the integrand significantly, especially when a composition of functions is involved.
Method 2
Integration by Parts
If the integrand is the product of two functions, say $ f(x) \cdot g'(x) $, this technique applies the product rule in reverse: $$ \int f(x)g'(x) \: dx = f(x)g(x) - \int f'(x)g(x) \: dx $$ It's particularly useful when differentiating one part of the product simplifies it.
Method 3
When the integrand involves the product of a function raised to a power and its derivative, the following shortcut can be used: $$ \int f'(x) \cdot [f(x)]^n \: dx = \frac{[f(x)]^{n+1}}{n+1} \quad \text{(for } n \ne -1 \text{)} $$
Example. To evaluate: $$ \int \cos(x) \cdot \sin^2(x) \: dx $$ Let $ f(x) = \sin(x) $, so $ f'(x) = \cos(x) $, and $ n = 2 $: $$ \int f'(x) \cdot [f(x)]^n = \frac{[\sin(x)]^{3}}{3} $$
Method 4
Partial Fraction Decomposition
When the integrand is a rational function $ \frac{P(x)}{Q(x)} $, it can often be broken down into simpler fractions: $$ \int \frac{P(x)}{Q(x)} \: dx = \int \left( \frac{A}{C(x)} + \frac{B}{D(x)} \right) \: dx = \int \frac{A}{C(x)} \: dx + \int \frac{B}{D(x)} \: dx $$ This is especially effective when the denominator can be factored into linear or irreducible quadratic terms.

And so on - each method unlocks a different class of integrals, and mastering them is essential for solving more complex problems.