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

Step by step

The question

Question: find the integral of sin² x cos³ x.

∫ sin² x cos³ x dx =

1 Save one

The power of cos is 3: odd.

Keep one cos x aside.

It will be part of du.

= ∫ cos² x sin² x cos x dx

2 Change the rest

The rule: sin squared plus cos squared is 1.

sin² x + cos² x = 1

So cos² x = 1 − sin² x.

Now everything else is sin x.

= ∫ (1 − sin² x) sin² x cos x dx

3 Choose u

Let u = sin x.

Its derivative is cos x: du = cos x dx.

That is the cos x dx we kept.

u = sin x, du = cos x dx

4 Write it with u

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

= ∫ (1 − u²)u² du

Multiply out the bracket.

= ∫ (u² − u⁴) du

Careful!

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

Save one and swap the rest.

Careful: not ∫ cos³ x dx = cos⁴ x4 ✗

5 Integrate

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

= u³3 − u⁵5 + C

6 Put x back

Put sin x back in place of u.

So the answer is sin³ x/3 − sin⁵ x/5 + C.

= sin³ x3 − sin⁵ x5 + C

7 Check

Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.112.

F(1.001) − F(0.999)0.002 ≈ 0.112

The question at x = 1 gives 0.112.

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

Right!

sin² 1 cos³ 1 ≈ 0.112

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