The integral of sin^5x is equal to ∫sin^{5}x dx = -(cos^{5}x)/5 + (2cos^{3}x)/3 -cos x+C, where C denotes an integration constant. Here we will learn how to integrate sin^5x.

The integration formula of sin^{5}x is given by

$\int \sin^5x$ $=-\dfrac{\cos^5x}{5}+\dfrac{2}{3} \cos^3x -\cos x +C$

where C is a constant.

## Integration of sin^{5}x

**Question:** Find the integration of sin^{5}x.

**Answer:**

We have:

∫sin^{5}x dx = ∫sin^{4}x sinx dx

= ∫(1-cos^{2}x)^{2} sinx dx as we know sin^{2}x = 1-cos^{2}x.

[Put z=cosx, so dz=-sinx dx]

= – ∫(1-z^{2})^{2} dz

= -∫(1-2z^{2} +z^{4}) dz

= – ∫dz + ∫2z^{2} dz + ∫z^{4} dz

= -z + 2z^{3}/3 + z^{5}/5 +C, by the power rule of integration ∫x^{n}dx = x^{n+1}/(n+1).

= $-\dfrac{\cos^5x}{5}+\dfrac{2}{3} \cos^3x -\cos x +C$

So the integration of sin^5x is equal to ∫sin^{5}x dx = -(cos^{5}x)/5 + (2cos^{3}x)/3 -cos x+C where C denotes an arbitrary integral constant.

**More Integrals:** How to integrate sin^{3}x?

## FAQs

### Q1: What is the integral of sin^5x?

**Answer:** The integral of sin^5x is given by ∫sin^{5}x dx = -(cos^{5}x)/5 + (2cos^{3}x)/3 -cos x+C where C is a constant.