Worked examples
The question
Question: find the integral of 1 over √(9 − x²).
∫ 1√9 − x2 dx =
1 Choose the substitution
We have the root of 9 minus x².
Use x = 3 sin θ, because 1 − sin² θ is cos² θ: the minus goes away.
√a² − x²: x = a sin θ
a² + x²: x = a tan θ
So x = 3 sin θ.
θ is an angle.
x = 3 sin θ
Its derivative: dx = 3 cos θ dθ.
dx = 3 cos θ dθ
Careful!
dx does not just become dθ.
dx is 3 cos θ dθ.
Careful: not dx = dθ ✗
2 Simplify the root
Put x = 3 sin θ in: x² is 9 sin² θ.
√9 − x2 = √9 − 9sin2 θ
Take out 9: its root is 3.
= 3√1 − sin2 θ
1 − sin² θ is cos² θ, and its root is cos θ.
= 3 cos θ
1 − sin2 θ = cos2 θ
3 Write it with θ
dx on top, the root at the bottom.
They are the same: they cancel!
= ∫ 3 cos θ3 cos θ dθ
= ∫ 1 dθ
4 Integrate
The integral of 1 is θ.
= θ + C
5 Back to x
From x = 3 sin θ: sin θ is x/3.
sin θ = x3
So θ is the angle whose sin is x/3: sin to the minus 1.
θ = sin−1(x/3)
So the answer is sin⁻¹(x/3) + C.
= sin−1(x3) + C
6 Check
Let's check: the slope of our answer F near x = 1, with a calculator: about 0.3536.
F(1.001) − F(0.999)0.002 ≈ 0.3536
The question at x = 1 gives 0.3536.
The same.
Right!
the question at x = 1: 1√9 − x2 ≈ 0.3536
Let's look back.
First: choose the substitution.
Then: simplify the root.
Then: write it with θ.
Then: integrate.
Then: back to x.
Last: check.
▶ Watch it solved
The question
Question: find the integral of 1/(x²√(4 − x²)).
∫ 1x2√4 − x2 dx =
1 Choose the substitution
We have the root of 4 minus x².
Use x = 2 sin θ, because 1 − sin² θ is cos² θ: the minus goes away.
√a² − x²: x = a sin θ
a² + x²: x = a tan θ
So x = 2 sin θ.
θ is an angle.
x = 2 sin θ
Its derivative: dx = 2 cos θ dθ.
dx = 2 cos θ dθ
Careful!
dx does not just become dθ.
dx is 2 cos θ dθ.
Careful: not dx = dθ ✗
2 Simplify the root
Put x = 2 sin θ in: x² is 4 sin² θ.
√4 − x2 = √4 − 4sin2 θ
Take out 4: its root is 2.
= 2√1 − sin2 θ
1 − sin² θ is cos² θ, and its root is cos θ.
= 2 cos θ
1 − sin2 θ = cos2 θ
3 Write it with θ
x² is 4 sin² θ, and the root is 2 cos θ.
2 cos θ cancels top and bottom.
= ∫ 2 cos θ4 sin2 θ × 2 cos θ dθ
= ∫ 14 sin2 θ dθ
1 over sin² θ is csc² θ.
The 1/4 waits in front.
= 14∫ csc2 θ dθ
4 Integrate
The derivative of cot θ is minus csc² θ.
So the integral of csc² θ is minus cot θ.
= −14cot θ + C
5 Back to x
Draw a right triangle for sin θ = x/2: opposite x, hypotenuse 2.
Pythagoras gives the third side: the root of 4 − x².
cot θ is adjacent over opposite.
cot θ = √4 − x2x
So the answer is minus the root of 4 − x², over 4x, + C.
= −√4 − x24x + C
6 Check
Let's check: the slope of our answer F near x = 1, with a calculator: about 0.5774.
F(1.001) − F(0.999)0.002 ≈ 0.5774
The question at x = 1 gives 0.5774.
The same.
Right!
the question at x = 1: 1x2√4 − x2 ≈ 0.5774
Let's look back.
First: choose the substitution.
Then: simplify the root.
Then: write it with θ.
Then: integrate.
Then: back to x.
Last: check.
▶ Watch it solved
The question
Question: find the integral of 1/(x²√(25 − x²)).
∫ 1x2√25 − x2 dx =
1 Choose the substitution
We have the root of 25 minus x².
Use x = 5 sin θ, because 1 − sin² θ is cos² θ: the minus goes away.
√a² − x²: x = a sin θ
a² + x²: x = a tan θ
So x = 5 sin θ.
θ is an angle.
x = 5 sin θ
Its derivative: dx = 5 cos θ dθ.
dx = 5 cos θ dθ
Careful!
dx does not just become dθ.
dx is 5 cos θ dθ.
Careful: not dx = dθ ✗
2 Simplify the root
Put x = 5 sin θ in: x² is 25 sin² θ.
√25 − x2 = √25 − 25sin2 θ
Take out 25: its root is 5.
= 5√1 − sin2 θ
1 − sin² θ is cos² θ, and its root is cos θ.
= 5 cos θ
1 − sin2 θ = cos2 θ
3 Write it with θ
x² is 25 sin² θ, and the root is 5 cos θ.
5 cos θ cancels top and bottom.
= ∫ 5 cos θ25 sin2 θ × 5 cos θ dθ
= ∫ 125 sin2 θ dθ
1 over sin² θ is csc² θ.
The 1/25 waits in front.
= 125∫ csc2 θ dθ
4 Integrate
The derivative of cot θ is minus csc² θ.
So the integral of csc² θ is minus cot θ.
= −125cot θ + C
5 Back to x
Draw a right triangle for sin θ = x/5: opposite x, hypotenuse 5.
Pythagoras gives the third side: the root of 25 − x².
cot θ is adjacent over opposite.
cot θ = √25 − x2x
So the answer is minus the root of 25 − x², over 25x, + C.
= −√25 − x225x + C
6 Check
Let's check: the slope of our answer F near x = 1, with a calculator: about 0.2041.
F(1.001) − F(0.999)0.002 ≈ 0.2041
The question at x = 1 gives 0.2041.
The same.
Right!
the question at x = 1: 1x2√25 − x2 ≈ 0.2041
Let's look back.
First: choose the substitution.
Then: simplify the root.
Then: write it with θ.
Then: integrate.
Then: back to x.
Last: check.
Answer = −√25 − x225x + C
▶ Watch it solved
