Free Printable Integration by Parts

1.

Evaluate the indefinite integral using integration by parts.
$\int3x\ e^{2x}\ dx\$

a)

$-\frac{xe^{2x}}{2}+\frac{e^{2x}}{4}+C$

b)

$\frac{3xe^{2x}}{2}-\frac{3e^{2x}}{4}+C$

c)

$xe^{-2x}+\frac{\left(1-x^2\right)^{ }}{2}+C$

d)

$-\frac{xe^{2x}}{2}+\frac{\ln e^{2x}}{4}+C$

2.

Evaluate the indefinite integral using integration by parts.
$\int t^2\ln t\ dt\$

a)

$\frac{2t^2\ln2t-t^2}{4}+C$

b)

$\frac{t^3\ \ln3}{3}-\frac{t^3}{9}+C$

c)

$\frac{e^t}{2t+2}+C$

d)

$\frac{-2t-1}{4e^{2t}}+C$

3.

$\int x\sin\left(8x\right)dx$

a)

$=-\frac{x}{8}\cos\left(8x\right)+\frac{1}{64}\sin\left(8x\right)+C$

b)

$=-\frac{x}{8}\cos\left(8x\right)-\frac{1}{64}\sin\left(8x\right)+C$

c)

$=-\frac{x}{8}\cos\left(8x\right)+\frac{1}{8}\sin\left(8x\right)+C$

d)

$=\frac{x}{8}\cos\left(8x\right)+\frac{1}{64}\sin\left(8x\right)+C$

4.

$\int xe^{-x}dx$

a)

$=-e^{-x}\left(x-1\right)+C$

b)

$=e^{-x}\left(x+1\right)+C$

c)

$=e^{-x}\left(x-1\right)+C$

d)

$=-e^{-x}\left(x+1\right)+C$

5.

Evaluate the indefinite integral using integration by parts.
$\int\log_5t\ dt\ ;\ \ \$

a)

$\frac{2t^2\ln3t-t^2}{4}+C$

b)

$-\frac{t}{3^t\ln3}-\frac{1}{3^t\cdot\left(\ln3\right)^2}+C$

c)

$t\ln\left(t^2+9\right)-2t+6\tan^{-1}\left(\frac{t}{3}\right)+C$

d)

$t\log_5t-\frac{t}{\ln5}+C$

6.

$\int_{ }x^2\sin x\ dx$ by using integration by parts.

a)

$x\sin x-\ e^x+C$

b)

$\left(\frac{-e^x\cos x-e^x\sin x+}{2}\right)\ +C$

c)

$xe^x+C$

d)

$xe^x-e^x+C$

7.

Evaluate the indefinite integral using integration by parts
$\int xe^{4x}dx\ ;\ \$

a)

$-\frac{x}{4^x\ln4}-\frac{1}{4^x\cdot\left(\ln4\right)^2}+C$

b)

$\frac{x^5\ln x}{5}-\frac{x^5}{25}+C$

c)

$x\ln\left(x+4\right)-x+4\ln\left(x+4\right)+C$

d)

$\frac{xe^{4x}}{4}-\frac{e^{4x}}{16}+C$

8.

Evaluate the indefinite integral using integration by parts
$\int e^{4x}\cos\left(2x\right)dx\$

a)

$\frac{e^{4x}\sin\left(2x\right)}{10}+\frac{e^{4x}\cos\left(2x\right)}{5}+C$

b)

$-\frac{e^{4x}\sin\left(2x\right)}{10}+\frac{e^{4x}\cos\left(2x\right)}{5}+C$

c)

$\frac{e^{4x}\cos\left(2x\right)}{10}+\frac{e^{4x}\sin\left(2x\right)}{5}+C$

d)

$\frac{e^{4x}\sin^2\left(2x\right)}{10}+C$

9.

Evaluate the indefinite integral using integration by parts
$\int e^{\sqrt{2x}}dx$

a)

$\sqrt{2}e^{\sqrt{2}\sqrt{x}}\sqrt{x}+e^{\sqrt{2}\sqrt{x}}+C$

b)

$2e^{\sqrt{2}\sqrt{x}}\sqrt{x}+C$

c)

$e^{\sqrt{2}\sqrt{x}}x-e^{\sqrt{2}\sqrt{x}}+C$

d)

$\sqrt{2}e^{\sqrt{2}\sqrt{x}}\sqrt{x}-e^{\sqrt{2}\sqrt{x}}+C$

10.

Evaluate the indefinite integral using integration by parts
$\int\sin\left(\sqrt{x}\right)dx$

a)

$-\sqrt{2}\sqrt{x}\sin^2\left(\sqrt{x}\right)+C$

b)

$-2\sqrt{x}\cos\left(\sqrt{x}\right)+2\sin\left(\sqrt{x}\right)+C$

c)

$\sqrt{2}\sqrt{x}\cos\left(\sqrt{x}\right)+\sin\sqrt{x}+C$

d)

$\sqrt{2}\left(-\sqrt{x}\cos\left(\sqrt{x}\right)-\sin\left(\sqrt{x}\right)\right)+C$

Integration by Parts

Total questions: 10

Worksheet time: 30mins

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