# How to Integrate sin^5x | Integral of sin^5x

The integral of sin^5x is equal to ∫sin5x dx = -(cos5x)/5 + (2cos3x)/3 -cos x+C, where C denotes an integration constant. Here we will learn how to integrate sin^5x.

The integration formula of sin5x 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 sin5x

Question: Find the integration of sin5x.

We have:

∫sin5x dx = ∫sin4x sinx dx

= ∫(1-cos2x)2 sinx dx as we know sin2x = 1-cos2x.

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

= – ∫(1-z2)2 dz

= -∫(1-2z2 +z4) dz

= – ∫dz + ∫2z2 dz + ∫z4 dz

= -z + 2z3/3 + z5/5 +C, by the power rule of integration ∫xndx = xn+1/(n+1).

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

So the integration of sin^5x is equal to ∫sin5x dx = -(cos5x)/5 + (2cos3x)/3 -cos x+C where C denotes an arbitrary integral constant.

More Integrals: How to integrate sin3x?

Integration of tan2x

Integral of e2x

## FAQs

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

Answer: The integral of sin^5x is given by ∫sin5x dx = -(cos5x)/5 + (2cos3x)/3 -cos x+C where C is a constant.