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Calculus II · Techniques of Integration

Integration by Parts — Practice Problems and Worked Solutions

Reverse the product rule with a strategic u and dv.

The formulas you need here

Integration by parts

\int u d v = uv - \int v d u

What it says: Trade one integral for another, hoping the new one is easier than the old one.
When to reach for it: A product of two unlike things — a polynomial times a trig, exponential or log.
Watch out: Choose u by LIATE (log, inverse trig, algebraic, trig, exponential): the earlier type becomes u. Picking the wrong u makes the new integral worse, not better.

\int x e^{x} d x = x e^{x} - e^{x} + C

u = x because differentiating it removes the x entirely.

16 worked problems

Every step is generated and checked by a computer algebra system.

1.
Evaluate using integration by parts.
$\int x e^{x} d x$
Answer: $(x - 1) e^{x} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x e^{x} d x = (x - 1) e^{x} + C$
2.
Evaluate using integration by parts.
$\int x sin (x) d x$
Answer: $- x cos (x) + sin (x) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x sin (x) d x = - x cos (x) + sin (x) + C$
3.
Evaluate using integration by parts.
$\int x cos (x) d x$
Answer: $x sin (x) + cos (x) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x cos (x) d x = x sin (x) + cos (x) + C$
4.
Evaluate using integration by parts.
$\int x^{2} e^{x} d x$
Answer: $(x^{2} - 2 x + 2) e^{x} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x^{2} e^{x} d x = (x^{2} - 2 x + 2) e^{x} + C$
5.
Evaluate using integration by parts.
$\int lo g (x) d x$
Answer: $x (lo g (x) - 1) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int lo g (x) d x = x lo g (x) - x + C$
6.
Evaluate using integration by parts.
$\int x lo g (x) d x$
Answer: $\frac{x ^{2} ( 2 l o g ( x ) - 1 )}{4} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x lo g (x) d x = \frac{x ^{2} lo g ( x )}{2} - \frac{x ^{2}}{4} + C$
7.
Evaluate using integration by parts.
$\int x e^{2 x} d x$
Answer: $\frac{( 2 x - 1 ) e ^{2 x}}{4} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x e^{2 x} d x = \frac{( 2 x - 1 ) e ^{2 x}}{4} + C$
8.
Evaluate using integration by parts.
$\int x sin (2 x) d x$
Answer: $- \frac{x c o s ( 2 x )}{2} + \frac{s i n ( 2 x )}{4} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x sin (2 x) d x = - \frac{x cos ( 2 x )}{2} + \frac{sin ( 2 x )}{4} + C$
9.
Evaluate using integration by parts.
$\int x cos (3 x) d x$
Answer: $\frac{x s i n ( 3 x )}{3} + \frac{c o s ( 3 x )}{9} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x cos (3 x) d x = \frac{x sin ( 3 x )}{3} + \frac{cos ( 3 x )}{9} + C$
10.
Evaluate using integration by parts.
$\int x^{2} sin (x) d x$
Answer: $- x^{2} cos (x) + 2 x sin (x) + 2 cos (x) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x^{2} sin (x) d x = - x^{2} cos (x) + 2 x sin (x) + 2 cos (x) + C$
11.
Evaluate using integration by parts.
$\int x^{2} cos (x) d x$
Answer: $x^{2} sin (x) + 2 x cos (x) - 2 sin (x) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x^{2} cos (x) d x = x^{2} sin (x) + 2 x cos (x) - 2 sin (x) + C$
12.
Evaluate using integration by parts.
$\int lo g (2 x) d x$
Answer: $x (lo g (2 x) - 1) + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int lo g (2 x) d x = x lo g (2 x) - x + C$
13.
Evaluate using integration by parts.
$\int x^{2} lo g (x) d x$
Answer: $\frac{x ^{3} ( 3 l o g ( x ) - 1 )}{9} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x^{2} lo g (x) d x = \frac{x ^{3} lo g ( x )}{3} - \frac{x ^{3}}{9} + C$
14.
Evaluate using integration by parts.
$\int x e^{- x} d x$
Answer: $(- x - 1) e^{- x} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x e^{- x} d x = (- x - 1) e^{- x} + C$
15.
Evaluate using integration by parts.
$\int atan (x) d x$
Answer: $x atan (x) - \frac{l o g ( x ^{2} + 1 )}{2} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int atan (x) d x = x atan (x) - \frac{lo g ( x ^{2} + 1 )}{2} + C$
16.
Evaluate using integration by parts.
$\int x^{3} e^{x} d x$
Answer: $(x^{3} - 3 x^{2} + 6 x - 6) e^{x} + C$
Step-by-step solution
1. $\int u d v = uv - \int v d u$
2. $\int x^{3} e^{x} d x = (x^{3} - 3 x^{2} + 6 x - 6) e^{x} + C$

The formulas you need here

Integration by parts

16 worked problems

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