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.

Answer:

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 e^2x

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