Integral of sin^n(x) Using the Integration by Parts Method

In calculus, integrating functions of the form sin^n(x) can be approached through various techniques, with the integration by parts method being particularly useful for handling even and odd powers of sine. This article explores how to find the integral of sin^n(x) using the integration by parts method, providing a detailed derivation and a recursive formula.

Step-by-Step Derivation

Step 1: Integration by Parts

The integral of sin^n(x) dx can be simplified using the integration by parts formula:

int u dv uv - int v du

Let:

u sin^{n-1}(x) dv sin(x) dx

Then, we differentiate and integrate to find:

du (n-1) sin^{n-2}(x) cos(x) dx v -cos(x)

Applying integration by parts, we get:

I_n -sin^{n-1}(x) cos(x) (n-1) int sin^{n-2}(x) cos^2(x) dx

Step 2: Using the Pythagorean Identity

We use the Pythagorean identity to simplify the integral further:

cos^2(x) 1 - sin^2(x)

Substituting this identity, we have:

I_n -sin^{n-1}(x) cos(x) (n-1) int sin^{n-2}(x) (1 - sin^2(x)) dx

This simplifies to:

I_n -sin^{n-1}(x) cos(x) (n-1) left( int sin^{n-2}(x) dx - int sin^n(x) dx right)

Step 3: Rearranging the Equation

Rearrange the equation to isolate I_n :

I_n (n-1) I_n -sin^{n-1}(x) cos(x)

Multiplying both sides by frac{1}{n} , we obtain:

frac{n}{n} I_n frac{-sin^{n-1}(x) cos(x) (n-1)}{n}

Simplifying, we get:

I_n frac{-sin^{n-1}(x) cos(x) (n-1)}{n} I_{n-2}

Final Formula

The final recursive formula for the integral of sin^n(x) dx is:

I_n frac{-sin^{n-1}(x) cos(x) (n-1)}{n} I_{n-2}

Where:

I_0 x C (constant of integration) I_1 -cos(x) C

This recursive formula allows us to calculate I_n in terms of I_{n-2} , making it a powerful tool for evaluating integrals of sine to any positive integer power n .

Conclusion

Understanding the integration by parts method and the reduction formula for integrals of the form sin^n(x) dx is crucial for advanced calculus problems. By breaking down the integral step-by-step, we can derive a recursive formula that simplifies the process. This not only helps in solving complex definite integrals but also provides a deeper insight into the structure of these integrals.