Chapter 21 Multiple choice questions

. Using integration by parts, find:

y = \int x^{3} e^{5 x} d x

.

y = \frac{e^{5 x}}{5} (x^{3} - 3 x^{2}) + c

y = \frac{{2e}^{5 x}}{125} (5 x - 1) + c

y = \frac{{3e}^{5 x}}{25} (x^{2} - 2 x) + c

y = \frac{e^{5 x}}{625} (125 x^{3} - 75 x^{2} + 30 x - 6) + c

. Using integration by parts, find:

y = \int x^{2} \ln 3 x d x

.

y = \frac{x^{3}}{9} (\ln 3 x - 1) + c

y = \frac{x^{3}}{9} (3 \ln 3 x - 1) + c

y = \frac{x^{3}}{9} (\ln 3 x + 1) + c

y = \frac{x^{3}}{9} (- \ln 3 x + 1) + c

. Using integration by parts, find:

y = \int e^{4 x} \sin 3 x d x

.

y = \frac{e^{4 x}}{25} (4 \sin 3 x - 3 \cos 3 x) + c

y = 25 e^{4 x} (4 \sin 3 x - 3 \cos 3 x) + c

y = \frac{e^{4 x}}{25} (3 \cos 3 x - 4 \sin 3 x) + c

y = 25 e^{4 x} (3 \cos 3 x - 4 \sin 3 x) + c

. Using integration by substitution, find:

y = \int x^{7} {(2 x^{8} - 5)}^{6} d x

.

y = \frac{{(2 x^{8} - 5)}^{7}}{42} + c

y = \frac{{(2 x^{8} - 5)}^{7}}{112} + c

y = \frac{x^{8} {(2 x^{8} - 5)}^{7}}{42} + c

y = \frac{x^{8} {(2 x^{8} - 5)}^{7}}{112} + c

. Using integration by substitution, find:

y = \int \frac{x^{2}}{4 x^{3} - 7} d x

.

y = \ln 4 x - \frac{x^{3}}{14} + c

y = \frac{1}{4 x^{2} - 7 x} + c

y = \ln (4 x^{3}) - \frac{3 x^{3}}{7} + c

y = \frac{\ln (4 x^{3} - 7)}{12} + c

. Using integration by substitution, find:

y = \int \frac{\cos (\sqrt{x})}{\sqrt{x}} d x

.

y = \sin (\sqrt{x}) + c

y = 2 \sin (\sqrt{x}) + c

y = \frac{1}{\sin (\sqrt{x})} + c

y = \frac{2}{\sin (\sqrt{x})} + c

. Using integration by backward substitution, find:

y = \int \frac{x}{\sqrt{x + 5}} d x

.

y = \sqrt{(x + 5)} + c

y = \ln (\sqrt{(x + 5)}) + c

y = \frac{2}{3} \sqrt{{(x + 5)}^{3}} + c

y = \frac{2}{3} (\sqrt{{(x + 5)}^{3}} - 15 \sqrt{x + 5}) + c

. Calculate the following integral:

y = \int \frac{1}{x^{2} + 4} d x

.

y = \frac{1}{2} t a n (2 x) + c

y = \frac{1}{2} \arctan (\frac{x}{2}) + c

y = - \frac{1}{2} \cot (2 x) + c

y = \frac{1}{2} \ln (x^{2} + 4) + c

. Calculate the following integral:

y = \int_{0}^{\infty} x^{3} e^{- 4 x} d x

.

y = \frac{3}{128}

y = \frac{3}{256}

y = \frac{1}{2} \sqrt{\frac{π}{4}}

y = \frac{\sqrt{π}}{2}

. Calculate the following integral:

y = \int_{0}^{\frac{π}{2}} \sin (4 x) \sin (6 x) d x

.

y = \frac{π}{2}

y = \frac{3 π}{4}

y = \frac{π}{4}

y = 0

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