Integration by Substitution (Part 2)

Integration by Substitution (Part 2)

How to solve the integration by substitution problems (definite integration): examples and their solutions.

Example 1

Find the given integral. The integral, from 0 to pi/2, of [e^(sin x) cos x] dx

Just like solving indefinite integration by substitution,
change the variable.

Set sin x = t.
Then cos x dx = dt.

Derivative of sin x

Now, here's the difference between
the definite integration and the indefinite integration.

In definite integration,
you should also change the limits,
because the variable is changed.

Use t = sin x to change the limits.

If x = 0, then t = 0.
If x = π/2, then t = 1.

sin x = t
cos x dx = dt
[0, π/2] → [0, 1]

So (given) = ∫01 et dt.

Solve the integration.

Indefinite integration of ex

Example 2

Find the given integral. The integral, from e to e^4, of [dx/(x ln x)]

Set ln x = t.
Then 1/x dx = dt.

Derivative of ln x

Next, use t = ln x to change the limits.

If x = e, then t = 1.
If x = e4, then t = 4.

ln x = t
1/x dx dx = dt
[e, e4] → [1, 4]

So (given) = ∫14 1/t dt.

Solve the integration.

Indefinite integration of 1/x

Logarithms of powers