Integration by Parts (Part 2)

Integration by Parts (Part 2)

How to solve the integration by parts problems (definite integration): formula, example, and its solution.

Formula

[integral, from a to b, of uv' dx] = [uv, from a to b] - [integral, from a to b, of u'v dx]

You've learned this formula in part 1.

Integration by parts (Part 1)

In definite integration,
after using the integration by parts,
put the upper limit and the lower limit into the result.

That's it.

Example

Find the given integral. The integral, from 1 to e, of (ln x)^2 dx

Think (ln x)2 as (ln x)2⋅1.

Then use the integration by parts.

Set u = (ln x)2 and v' = 1.

Write u = (ln x)2.
Write u' = 2(ln x)(1/x).

Chain rule in differentiation

Derivative of ln x

Write v' = 1 next to u' = 2(ln x)(1/x).
And write v = x next to u = (ln x)2.

The integral [from 1 to e] of, uv', (ln x)2⋅1 dx
is equal to,
uv, (ln x)2x [from 1 to e]
minus the integral [from 1 to e] of, u'v, 2(ln x)(1/x)⋅x dx.

Put e and 1 into [(ln x)2x].

In 2(ln x)(1/x)⋅x,
cancel (1/x) and x.

Solve the definite integral.

Indefinite integration of ln x