Worked examples
The question
Question: find the integral of sin² x cos x.
∫ sin2 x cos x dx =
1 Choose u
Let u = sin x.
Its derivative is cos x: du = cos x dx.
And cos x dx is right there in the integral.
u = sin x, du = cos x dx
2 Write it with u
sin x becomes u, and cos x dx becomes du.
= ∫ u2 du
Careful!
cos x dx becomes du.
It does not stay in the integral as well.
Careful: cos x dx → du: it is used up
3 Integrate
The power rule: add 1 to each power, divide by the new power.
= u33 + C
4 Put x back
Put sin x back in place of u.
So the answer is sin³ x/3 + C.
= sin3 x3 + C
5 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.383.
F(1.001) − F(0.999)0.002 ≈ 0.383
The question at x = 1 gives 0.383.
The same: the slope of the answer is the question.
Right!
sin² 1 cos 1 ≈ 0.383
Let's look back.
First: choose u.
Then: write it with u.
Then: integrate.
Then: put x back.
Last: check.
▶ Watch it solved
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 = cos5 x5 − cos3 x3 + C
▶ Watch it solved
The question
Question: find the integral of cos⁵ x.
∫ cos5 x dx =
1 Save one
The power of cos is 5: odd.
Keep one cos x aside.
It will be part of du.
= ∫ cos4 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)2 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)2 du
Multiply out: (1 − u²)² is 1 − 2u² + u⁴.
Then times the rest.
= ∫ (u4 − 2u2 + 1) 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 ∫ cos5 x dx = cos6 x6 ✗
5 Integrate
The power rule: add 1 to each power, divide by the new power.
= u55 − 2u33 + u + C
6 Put x back
Put sin x back in place of u.
So the answer is sin⁵ x/5 − 2sin³ x/3 + sin x + C.
= sin5 x5 − 2sin3 x3 + sin x + C
7 Check
Let's check: the slope of our answer F near x = 1, with a calculator in radians: about 0.046.
F(1.001) − F(0.999)0.002 ≈ 0.046
The question at x = 1 gives 0.046.
The same: the slope of the answer is the question.
Right!
cos⁵ 1 ≈ 0.046
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 = sin5 x5 − 2sin3 x3 + sin x + C
▶ Watch it solved
