Integrate ∫ sin^5 x dx Step by Step

Step by step

The question

Question: find the integral of sin⁵ x.

∫ sin⁵ x dx =

1 Save one

The power of sin is 5: odd.

Keep one sin x aside.

It will be part of du.

= ∫ sin⁴ 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)² 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)² du

Multiply out: (1 − u²)² is 1 − 2u² + u⁴.

Then times the rest.

= ∫ (2u² − u⁴ − 1) 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 ∫ sin⁵ x dx = sin⁶ x6 ✗

5 Integrate

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

= 2u³3 − u⁵5 − u + C

6 Put x back

Put cos x back in place of u.

So the answer is 2cos³ x/3 − cos⁵ x/5 − cos x + C.

= 2cos³ x3 − cos⁵ x5 − cos x + C

7 Check

Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.422.

F(1.001) − F(0.999)0.002 ≈ 0.422

The question at x = 1 gives 0.422.

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

Right!

sin⁵ 1 ≈ 0.422

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