Integration by Substitution (Part 1)

Integration by Substitution (Part 1)

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

Example 1

Find the given indefinite integral. The integral of (2x - 1)^8 dx

Set 2x - 1 = t.

Differentiate both sides:
2 dx = dt
dx = (1/2)dt.

Implicit differentiation

Substitute [2x - 1] with [t].
And substitute [dx] with [(1/2)dt].

Then (given) = ∫ t8 (1/2)dt.

Power rule in integration

Put 2x - 1 in t.

Then (given) = (1/18)(2x - 1)9 + C.

Example 2

Find the given indefinite integral. The integral of (sin^5 x cos x) dx

Set sin x = t.

Differentiate both sides:
cos x dx = dt.

Implicit differentiation

Derivative of sin x

Substitute [sin x] with [t].
And substitute [cos x dx] with [dt].

Then (given) = ∫ t5 dt.

Power rule in integration

Put sin x in t.

Then (given) = (1/6) sin6 x + C.