Integrate ∫ sin^3 x cos^2 x dx Step by Step

Step by step

The question

Question: find the integral of sin³ x cos² x.

∫ sin³ x cos² x dx =

1 Save one

The power of sin is 3: odd.

Keep one sin x aside.

It will be part of du.

= ∫ sin² x cos² x sin x dx

2 Change the rest

The rule: sin squared plus cos squared is 1.

sin² x + cos² x = 1

So sin² x = 1 − cos² x.

Now everything else is cos x.

= ∫ (1 − cos² x) cos² x sin x dx

3 Choose u

Let u = cos x.

Its derivative is minus sin x: du = −sin x dx.

So sin x dx is −du.

u = cos x, du = −sin x dx

4 Write it with u

cos x becomes u, and sin x dx becomes −du.

= ∫ −(1 − u²)u² du

Multiply out the bracket.

= ∫ (u⁴ − u²) du

Careful!

The power rule does not work here: the derivative of sin x would have to be there too.

Save one and swap the rest.

Careful: not ∫ sin³ x dx = sin⁴ x4 ✗

5 Integrate

The power rule: add 1 to each power, divide by the new power.

= u⁵5 − u³3 + C

6 Put x back

Put cos x back in place of u.

So the answer is cos⁵ x/5 − cos³ x/3 + C.

= cos⁵ x5 − cos³ x3 + C

7 Check

Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.174.

F(1.001) − F(0.999)0.002 ≈ 0.174

The question at x = 1 gives 0.174.

The same: the slope of the answer is the question.

Right!

sin³ 1 cos² 1 ≈ 0.174

Let's look back.

First: save one.

Then: change the rest.

Then: choose u.

Then: write it with u.

Then: integrate.

Then: put x back.

Last: check.

Answer