# Test: Integration I - Challenging

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Question 1:   Solve integral$\int \left(\frac{1}{\sqrt{1-{\mathrm{sin}}^{2}\left(3x\right)}}\right)d\left(\mathrm{sin}\left(3x\right)\right)$
$F\left(x\right)=\mathrm{sin}\left(3x\right)+C$
$F\left(x\right)=\mathrm{arcsin}\left(\mathrm{sin}\left(3x\right)\right)+C$
$F\left(x\right)=sin\left(\mathrm{arcsin}\left(3x\right)\right)+C$
$F\left(x\right)=\mathrm{arcsin}\left(\mathrm{sin}\left(3x\right)\right)$
Question 2:   Find the antiderivative of the function $f\left(x\right)={\mathrm{sin}}^{2}\left(5x\right)$
$F\left(x\right)=\frac{1}{2}\left(x-\mathrm{sin}\left(10x\right)\right)+C$
$F\left(x\right)=x-\frac{1}{10}×\mathrm{sin}\left(10x\right)+C$
$F\left(x\right)=\frac{1}{2}\left(x-\frac{1}{10}×\mathrm{sin}\left(10x\right)\right)+C$
$F\left(x\right)=\frac{{\mathrm{sin}}^{3}\left(5x\right)}{3}+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\frac{1}{1+5{x}^{2}}$
$F\left(x\right)=\sqrt{5}\mathrm{arccot}\left(\sqrt{5}x\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{5}}\mathrm{arccot}\left(x\right)+C$
$F\left(x\right)=5\mathrm{arccot}\left(\sqrt{5}x\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{5}}\mathrm{arccot}\left(\sqrt{5}x\right)+C$
Question 4:   Calculate $\underset{0}{\overset{\pi }{\int }}\left(\mathrm{cos}\left(\frac{x}{2}\right)\right)dx$
$1$
$4$
$2$
$\pi$
Question 5:   Find the antiderivative of the function $f\left(x\right)=8{x}^{3}+3{x}^{2}$ that passes via point $\left(1;5\right)$
$F\left(x\right)=2{x}^{4}+{x}^{3}+C$
$F\left(x\right)=2{x}^{4}+{x}^{3}$
$F\left(x\right)=2{x}^{4}+{x}^{3}+2$
$F\left(x\right)=2{x}^{4}+{x}^{3}+5$
Question 6:   Find the antiderivative of the function $f\left(x\right)={e}^{x}+3$ hat passes via point $\left(0;-7\right)$
$F\left(x\right)={e}^{x}+3x+2$
$F\left(x\right)={e}^{x}+3x+C$
$F\left(x\right)={e}^{x}+3x+8$
$F\left(x\right)={e}^{x}+3x-8$
Question 7:   Calculate the area of the figure defined by $y=\sqrt{x},\text{\hspace{0.17em}}x=\frac{2}{3},\text{\hspace{0.17em}}x=\frac{8}{3}$
$2\pi$
$\frac{3\pi }{10}$
$\frac{10\pi }{3}$
$\frac{5\pi }{3}$
Question 8:   Find the formula for the area of the shaded region:

$S=\underset{0}{\overset{\pi }{\int }}\left(\mathrm{sin}\left(x\right)\right)dx$
$S=\underset{0}{\overset{\pi }{\int }}\left(\mathrm{sin}\left(x\right)\right)dx-\underset{\pi }{\overset{2\pi }{\int }}\left(\mathrm{sin}\left(x\right)\right)dx$
$S=-\underset{0}{\overset{2\pi }{\int }}\left(\mathrm{sin}\left(x\right)\right)dx$
$S=\underset{0}{\overset{2\pi }{\int }}\left(\mathrm{sin}\left(x\right)\right)dx$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=3x$
$5$
$9$
$3$
$4.5$
Question 10:   Calculate the area of the figure defined by $y=\frac{x}{2}$, $x=4$ and $y=0$
$6$
$7$
$2$
$4$