Test: Integration II - Challenging

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Question 1:   Solve integral $\int \left(\frac{1}{{\mathrm{cos}}^{2}\left({e}^{x}\right)}\right)d\left({e}^{x}\right)$
$F\left(x\right)=\mathrm{tan}\left(x\right)+C$
$F\left(x\right)={e}^{x}+C$
$F\left(x\right)=\mathrm{tan}\left({e}^{x}\right)$
$F\left(x\right)=\mathrm{tan}\left({e}^{x}\right)+C$
Question 2:   Find the antiderivative of the function $f\left(x\right)={\mathrm{cos}}^{2}\left(3x\right)$
$F\left(x\right)=x+\frac{1}{6}×\mathrm{sin}\left(6x\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x+\mathrm{sin}\left(6x\right)\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x+\mathrm{sin}\left(x\right)\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x+\frac{1}{6}×\mathrm{sin}\left(6x\right)\right)+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\frac{5}{1+16{x}^{2}}$
$F\left(x\right)=\frac{1}{4}\mathrm{arctan}\left(4x\right)+C$
$F\left(x\right)=5\mathrm{arctan}\left(4x\right)+C$
$F\left(x\right)=\frac{5}{4}\mathrm{arctan}\left(4x\right)+C$
$F\left(x\right)=\frac{4}{5}\mathrm{arctan}\left(4x\right)+C$
Question 4:   Calculate $\underset{0}{\overset{\pi }{\int }}\left(\mathrm{sin}\left(\frac{x}{3}\right)\right)dx$
$\pi$
$1.5$
$3.5$
$1$
Question 5:   Find the antiderivative of the function $f\left(x\right)=9{x}^{2}-x$ that passes via point $\left(1;1\right)$
$F\left(x\right)=3{x}^{3}-\frac{{x}^{2}}{2}-1$
$F\left(x\right)=3{x}^{3}-\frac{{x}^{2}}{2}+2$
$F\left(x\right)=3{x}^{3}-\frac{{x}^{2}}{2}+C$
$F\left(x\right)=3{x}^{3}-\frac{{x}^{2}}{2}-1.5$
Question 6:   Find the antiderivative of the function $f\left(x\right)=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)$ that passes via point $\left(\pi ;0\right)$
$F\left(x\right)=-\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)-1$
$F\left(x\right)=-\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)+1$
$F\left(x\right)=-\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)+C$
$F\left(x\right)=-\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)$
Question 7:   Calculate the area of the figure defined by $y=x,\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}x=9.$
$81\pi$
$234\pi$
$225\pi$
$729\pi$
Question 8:   Find the formula for the area of the shaded region:

$S=-\underset{0}{\overset{2\pi }{\int }}\left(\mathrm{cos}\left(x\right)\right)dx$
$S=\underset{0}{\overset{\pi }{\int }}\left(\mathrm{cos}\left(x\right)\right)dx$
$S=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\mathrm{cos}\left(x\right)\right)dx-\underset{\frac{\pi }{2}}{\overset{\frac{3\pi }{2}}{\int }}\left(\mathrm{cos}\left(x\right)\right)dx+\underset{\frac{3\pi }{2}}{\overset{2\pi }{\int }}\left(\mathrm{cos}\left(x\right)\right)dx$
$S=\underset{0}{\overset{2\pi }{\int }}\left(\mathrm{cos}\left(x\right)\right)dx$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=2x$
$4$
$1$
$\frac{3}{4}$
$\frac{4}{3}$
Question 10:   Calculate the area of the figure defined by $y=3x$, $x=2$ and $y=0$
$4$
$7$
$2$
$6$