Knowing which integration method to use in different situations is important for solving hard integrals. Each method works best in certain cases.

Integration by Parts
This method is great when you have the product of two functions to integrate. It comes from the product rule of differentiation, which says:

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

When to use it:
You should use this method when you have things like polynomials multiplied by exponential functions or trigonometric functions. For example, with $x e^x$ or $x \sin(x)$. Choosing your $u$ and $dv$ wisely can make the integration easier.

Trigonometric Substitution
You use trigonometric substitution when you're integrating expressions that include square roots of certain types of equations. If the integral has square roots like $\sqrt{a^2 - x^2}$, $\sqrt{x^2 + a^2}$, or $\sqrt{x^2 - a^2}$, this method is helpful.

When to use it:
This method is useful for integrals that include roots or that seem tough to simplify. For example, to integrate $\int \sqrt{4 - x^2} \, dx$, you can let $x = 2 \sin(\theta)$, which makes the problem a lot easier.

Partial Fractions
This technique is key for integrating fractions where the top number (numerator) has a lower degree than the bottom number (denominator). By breaking down a complicated fraction into simpler parts, it makes integration easier.

When to use it:
You can use this method for integrals like $\int \frac{2}{x^2 - 1} \, dx$. First, factor the denominator, then use the right substitutions for each part in the partial fraction breakdown.

In Summary:

• Integration by Parts works best for products of functions.
• Trigonometric Substitution is good for integrals with square roots that relate to certain quadratics.
• Partial Fractions helps to simplify rational functions.

By looking at the structure of the integrand and picking the right method, you can handle advanced integration techniques with ease.