Welcome to this module on integrating trigonometric functions, a crucial skill for success in competitive exams like JEE. This section will equip you with the fundamental techniques and strategies to confidently tackle problems involving trigonometric integrals.

Many trigonometric integrals can be simplified by using fundamental trigonometric identities. Recognizing and applying these identities is key to transforming complex integrals into manageable forms.

Memorizing the basic integrals of trigonometric functions is fundamental. These form the building blocks for more complex integrations.

Beyond basic identities and standard integrals, several techniques are vital for solving more complex trigonometric integrals.

This is one of the most powerful techniques. Look for a part of the integrand whose derivative is also present (or can be made present with a constant factor). For example, in $\int \sin^3(x) \cos(x) dx$, let $u = \sin(x)$, then $du = \cos(x) dx$. The integral becomes $\int u^3 du$.

Use this when the integrand is a product of two functions, and substitution doesn't seem to work directly. The formula is $\int u dv = uv - \int v du$. Choosing 'u' and 'dv' wisely (often using the LIATE rule: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) is crucial.

These formulas express the integral of a trigonometric function with a higher power in terms of the integral of the same function with a lower power. For example, reduction formulas for $\int \sin^n(x) dx$ and $\int \cos^n(x) dx$ are invaluable for integrating higher powers.

As mentioned earlier, identities like $\cos^2(x) = \frac{1+\cos(2x)}{2}$ and $\sin^2(x) = \frac{1-\cos(2x)}{2}$ are essential for reducing powers of sine and cosine, making them integrable using basic formulas.

These integrals are solved using product-to-sum identities:

For rational functions of sine and cosine, the Weierstrass substitution ($t = \tan(x/2)$) is often employed. This transforms the trigonometric integral into an integral of a rational function in 't', which can then be solved using partial fractions or other standard techniques for rational functions. However, this can be computationally intensive, so always check if simpler methods apply first.

Practice is paramount. Work through a variety of problems, starting with simpler ones and gradually moving to more complex scenarios. Pay close attention to the structure of the integrand to determine the most efficient integration technique.

Let's integrate $\int \sin^3(x) \cos^2(x) dx$.

1. Rewrite $\sin^3(x)$ as $\sin^2(x) \sin(x)$.
2. Use the identity $\sin^2(x) = 1 - \cos^2(x)$. So the integral becomes $\int (1 - \cos^2(x)) \cos^2(x) \sin(x) dx$.
3. Let $u = \cos(x)$. Then $du = -\sin(x) dx$, which means $\sin(x) dx = -du$.
4. Substitute: $\int (1 - u^2) u^2 (-du) = -\int (u^2 - u^4) du$.
5. Integrate: $-(\frac{u^3}{3} - \frac{u^5}{5}) + C = \frac{u^5}{5} - \frac{u^3}{3} + C$.
6. Substitute back $u = \cos(x)$: $\frac{\cos^5(x)}{5} - \frac{\cos^3(x)}{3} + C$.