# Test: Integration III - Challenging

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Question 1:   Solve integral $\int \left({5}^{\mathrm{cos}\left(x\right)}\right)d\left(\mathrm{cos}\left(x\right)\right)$
$F\left(x\right)={5}^{\mathrm{cos}\left(x\right)}+C$
$F\left(x\right)=5\mathrm{cos}\left(x\right)+C$
$F\left(x\right)=\frac{{5}^{\mathrm{cos}\left(x\right)}}{\mathrm{ln}\left(5\right)}+C$
$F\left(x\right)=\frac{{5}^{\mathrm{cos}\left(x\right)}}{\mathrm{ln}\left(5\right)}$
Question 2:   Find the antiderivative of the function $f\left(x\right)={\mathrm{cos}}^{2}\left(7x\right)$
$F\left(x\right)=\frac{1}{2}\left(x+\mathrm{sin}\left(14x\right)\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x+\frac{1}{14}×\mathrm{sin}\left(14x\right)\right)+C$
$F\left(x\right)=x+\frac{1}{14}×\mathrm{sin}\left(14x\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x+\mathrm{sin}\left(x\right)\right)+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\frac{8}{1+9{x}^{2}}$
$F\left(x\right)=\frac{8}{3}\mathrm{arctan}\left(3x\right)+C$
$F\left(x\right)=8\mathrm{arctan}\left(3x\right)+C$
$F\left(x\right)=\frac{1}{3}\mathrm{arctan}\left(3x\right)+C$
$F\left(x\right)=\frac{8}{3}\mathrm{arctan}\left(\frac{x}{3}\right)+C$
Question 4:   Calculate $\underset{0}{\overset{\pi }{\int }}\left(\frac{1}{{\mathrm{cos}}^{2}\left(\frac{x}{4}\right)}\right)dx$
$5$
$4$
$\pi$
$-3$
Question 5:   Find the antiderivative of the function $f\left(x\right)=5x+1$ that passes via point $\left(2;5\right)$
$F\left(x\right)=\frac{5{x}^{2}}{2}+x$
$F\left(x\right)=\frac{5{x}^{2}}{2}+x-7$
$F\left(x\right)=\frac{5{x}^{2}}{2}+x+7$
$F\left(x\right)=\frac{5{x}^{2}}{2}+x-3$
Question 6:   Find the antiderivative of the function $f\left(x\right)={7}^{x}+2$ that passes via point $\left(0;0\right)$
$F\left(x\right)=\frac{{7}^{x}}{\mathrm{ln}\left(7\right)}+2x-\frac{1}{\mathrm{ln}\left(7\right)}$
$F\left(x\right)=\frac{{7}^{x}}{\mathrm{ln}\left(7\right)}+x-\frac{1}{\mathrm{ln}\left(7\right)}$
$F\left(x\right)=\frac{{7}^{x}}{\mathrm{ln}\left(7\right)}+2x+\frac{1}{\mathrm{ln}\left(7\right)}$
$F\left(x\right)=\frac{{7}^{x}}{\mathrm{ln}\left(7\right)}+2x+\frac{3}{\mathrm{ln}\left(7\right)}$
Question 7:   Calculate the area of the figure defined by $y=\sqrt{\mathrm{cos}\left(x\right)},\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}x=\frac{\pi }{6}$
$2\pi$
$\frac{5\pi }{2}$
$\frac{\pi }{2}$
$\frac{1}{2}$
Question 8:   Find the formula for the area of the shaded region:

$S=-\underset{0}{\overset{4}{\int }}\left(\sqrt{x}\right)dx$
$S=\underset{0}{\overset{4}{\int }}|\sqrt{x}|dx$
$S=\underset{0}{\overset{2}{\int }}\left(\sqrt{x}\right)dx$
$S=\underset{0}{\overset{4}{\int }}\left(\sqrt{x}\right)dx$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=x$
$\frac{1}{6}$
$6$
$1$
$\frac{1}{4}$
Question 10:   Calculate the area of the figure defined by $y=x$, $x=5$ and $y=0$
$12.5$
$12$
$10$
$13.5$