Integration by parts: choosing u, repeated parts

Warm-up

Three quick questions from earlier lessons.

Discover

For $\displaystyle\int x\ln x\,dx$ you could take $u = x$ or $u = \ln x$ . With $u = x$ you need $v = \displaystyle\int \ln x\,dx$ first. Which choice is easier?

$u = \ln x$
$u = x$

Key idea

Let $u$ be the factor that simplifies when differentiated.

Good

u

choices, in order: Logarithms, Inverse trig, Algebraic (powers of

x

), Trig, Exponentials (LIATE). With

x^n

times

e^x

or

\sin x

, apply parts

n

times.

Worked example

$\displaystyle\int x^2 e^x\,dx$ .

x^2 e^x - \displaystyle\int 2x e^x\,dx

u = x², v = eˣ.

x^2 e^x - (2xe^x - 2e^x)

Parts again on ∫2xeˣ: 2xeˣ − 2eˣ.

(x^2 - 2x + 2)e^x + C

Tidy.

Your turn, guided

Your turn: $\displaystyle\int x\ln x\,dx$ with $u = \ln x$ , $v = \tfrac{x^2}{2}$ .

\tfrac{x^2}{2}\ln x - \tfrac{x^2}{4} + C

(x²/2) ln x − ∫x/2 dx.

Practice ★

To integrate $\displaystyle\int x^{ 1 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 1 }\,dx$
2. $u = x^{ 1 }$ , $dv = \ln x\,dx$
3. $u = x^{ 1 }\ln x$ , $dv = dx$
To integrate $\displaystyle\int x^{ 2 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 2 }\,dx$
2. $u = x^{ 2 }$ , $dv = \ln x\,dx$
3. $u = x^{ 2 }\ln x$ , $dv = dx$

Evaluate $\displaystyle\int_0^1 x e^{ 2x}\,dx$ .
Evaluate $\displaystyle\int_0^1 x e^{ 5x}\,dx$ .

Watch out!

Practice ★★

Find $F$ with $F'(x) = 5x^2 e^x$ and $F(0) = 0$ .
Find $F$ with $F'(x) = 4x^2 e^x$ and $F(0) = 0$ .

Exit ticket

Three questions to check you have got it.

Evaluate $\displaystyle\int_0^1 x e^{ 2x}\,dx$ .
Evaluate $\displaystyle\int_0^1 x e^{ 5x}\,dx$ .
Evaluate $\displaystyle\int_0^1 x e^{ 3x}\,dx$ .

Summary card

Choosing $u$

LIATE: logs, inverse trig, powers, trig, exponentials.
$x^n e^x$ : parts $n$ times.

Watch out: if the new integral is harder, swap $u$ and $dv$ .

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