# Test: Integration I - Ambitious

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Question 1:   Find the antiderivative of the function $f\left(x\right)={\left(x-1\right)}^{2}$
$F\left(x\right)=\frac{{x}^{3}}{3}-\frac{{x}^{2}}{2}+x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}-{x}^{2}+x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}+x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}-{x}^{2}+x$
Question 2:   Find the antiderivative of the function $f\left(x\right)={\left({e}^{x}\right)}^{2}$
$F\left(x\right)=\frac{{e}^{2x}}{2}$
$F\left(x\right)=\frac{{e}^{2x}}{2}+C$
$F\left(x\right)=\frac{{e}^{x}}{2}+C$
$F\left(x\right)={e}^{2x}+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\frac{1}{2x}+4{x}^{3}$
$F\left(x\right)=\frac{1}{2}\mathrm{ln}|x|+\frac{{x}^{4}}{4}+C$
$F\left(x\right)=\frac{1}{2}\mathrm{ln}|x|+{x}^{4}+C$
$F\left(x\right)=\mathrm{ln}|x|+{x}^{4}+C$
$F\left(x\right)=\frac{1}{2}\mathrm{ln}|x|+{x}^{4}$
Question 4:   Find the antiderivative of the function $f\left(x\right)=2\mathrm{sin}\left(x\right)+7\mathrm{cos}\left(x\right)$
$F\left(x\right)=2\mathrm{cos}\left(x\right)+7\mathrm{sin}\left(x\right)+C$
$F\left(x\right)=-2\mathrm{cos}\left(x\right)+7\mathrm{sin}\left(x\right)$
$F\left(x\right)=-2\mathrm{cos}\left(x\right)-7\mathrm{sin}\left(x\right)+C$
$F\left(x\right)=-2\mathrm{cos}\left(x\right)+7\mathrm{sin}\left(x\right)+C$
Question 5:   Find the antiderivative of the function $f\left(x\right)=\frac{1}{16+{x}^{2}}$
$F\left(x\right)=\frac{1}{2}\mathrm{arctan}\left(\frac{x}{2}\right)+C$
$F\left(x\right)=\frac{1}{4}\mathrm{arctan}\left(x\right)+C$
$F\left(x\right)=\frac{1}{4}\mathrm{arctan}\left(\frac{x}{4}\right)+C$
$F\left(x\right)=\frac{1}{4}\mathrm{arctan}\left(\frac{x}{4}\right)$
Question 6:   Calculate $\underset{0}{\overset{\frac{2\pi }{3}}{\int }}\left(\mathrm{cos}\left(\frac{3x}{4}\right)\right)dx$
$1$
$0$
$\frac{4}{3}$
$\frac{3}{4}$
Question 7:   Calculate $\underset{0}{\overset{5}{\int }}\left({\left(\frac{3}{5}x\right)}^{2}-5x\right)dx$
$-47.5$
$50.5$
$-50.5$
$47.5$
Question 8:   Calculate $\underset{1}{\overset{2}{\int }}\left({2}^{x}-3\right)dx$
$\frac{2}{\mathrm{ln}\left(2\right)}$
$0$
$-1$
$\frac{2}{\mathrm{ln}\left(2\right)}-3$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=4$
$\frac{3}{32}$
$4$
$1$
$\frac{32}{3}$
Question 10:   Calculate the area of the figure defined by $y=2x-4$ , $x=5$ and $y=0$
$3$
$9$
$12$
$25$