The integration of sin^3x cos^2x is equal to (cos^{5}x)/5 – (cos^{3}x)/3 +C where C is a constant. Here we learn how to integrate sin^3x cos^2x.

## What is the Integration of sin^{3}x cos^{2}x?

**Question:** Find the integral of sin^{3}x cos^{2}x, that is,

Find ∫sin^{3}x cos^{2}x dx

**Solution:**

We have:

∫sin^{3}x cos^{2}x dx = ∫sin^{2}x sin x cos^{2}x dx

⇒ ∫sin^{3}x cos^{2}x dx = ∫(1 -cos^{2}x) cos^{2}x sin x dx [by the formula sin^{2}x = 1 -cos^{2}x]

Put cosx =z, so that -sinx dx =dz.

∴ From above, we have that

∫sin^{3}x cos^{2}x dx = – ∫(1 – z^{2}) z^{2} dz

= – ∫(z^{2} – z^{4}) dz

= – ∫z^{2} dz + ∫z^{4} dz

= – z^{3}/3 + z^{5}/5 +C

= – cos^{3}x/3 + cos^{5}x/5 +C

So the integration of sin^{3}x cos^{2}x equals to – cos^{3}x/3 + cos^{5}x/5 +C, that is, the integral of sin cube x cos square x is given by the formula:

∫sin^{3}x cos^{2}x dx = – cos^{3}x/3 + cos^{5}x/5 +C |

Related Integrals:

## FAQs

**Q1: What is the integral of sin^3x cos^2x?**

**Answer:** The integration of sin^3x cos^2x is given by ∫sin^{3}x cos^{2}x dx = – cos^{3}x/3 + cos^{5}x/5 +C where C denotes an integral constant.