# Test: Integration II - Ambitious

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Question 1:   Find the antiderivative of the function $f\left(x\right)={\left(2x+5\right)}^{2}$
$F\left(x\right)=\frac{4{x}^{3}}{3}+5{x}^{2}+25x$
$F\left(x\right)=\frac{4{x}^{3}}{3}+5{x}^{2}+25x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}+5{x}^{2}+25x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}+\frac{{x}^{2}}{2}+25+C$
Question 2:   Find the antiderivative of the function $f\left(x\right)={\left({2}^{6x}\right)}^{2}$
$F\left(x\right)={2}^{12x}+C$
$F\left(x\right)=\frac{{2}^{12x}}{12\mathrm{ln}\left(2\right)}$
$F\left(x\right)=\frac{{2}^{12x}}{12\mathrm{ln}\left(2\right)}+C$
$F\left(x\right)=\frac{{2}^{12x}}{\mathrm{ln}\left(2\right)}+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\frac{1}{{\mathrm{cos}}^{2}\left(2x\right)}$
$F\left(x\right)=\frac{1}{2}\mathrm{tan}\left(2x\right)$
$F\left(x\right)=\frac{1}{2}\mathrm{tan}\left(2x\right)+C$
$F\left(x\right)=\mathrm{tan}\left(2x\right)+C$
$F\left(x\right)=\frac{1}{2}\mathrm{tan}\left(x\right)+C$
Question 4:   Find the equation of integral for the area of the shaded region:

$S=\int \left(\left(x+1\right)-{x}^{2}\right)dx$
$S=\int \left(x+1\right)dx$
$S=\int \left({x}^{2}-\left(x+1\right)\right)dx$
$S=\int \left({x}^{2}\right)dx$
Question 5:   Find the antiderivative of the function $f\left(x\right)=\frac{1}{21+{x}^{2}}$
$F\left(x\right)=\frac{1}{\sqrt{21}}\mathrm{arctan}\left(\frac{x}{\sqrt{21}}\right)+C$
$F\left(x\right)=\mathrm{arctan}\left(\frac{x}{\sqrt{21}}\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{21}}\mathrm{arctan}\left(x\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{21}}\mathrm{arctan}\left(\frac{x}{\sqrt{21}}\right)$
Question 6:   Calculate $\underset{\pi }{\overset{2\pi }{\int }}\left(-\mathrm{sin}\left(\frac{x}{3}\right)\right)dx$
$-3$
$-\frac{1}{3}$
$1$
$0$
Question 7:   Calculate $\underset{0}{\overset{3}{\int }}\left({\left(\frac{4}{9}x\right)}^{2}-2\right)dx$
$\frac{38}{9}$
$-\frac{38}{9}$
$5$
$-\frac{38}{3}$
Question 8:   Calculate $\underset{0}{\overset{1}{\int }}\left({5}^{x}+1\right)dx$
$\frac{4}{\mathrm{ln}\left(5\right)}-1$
$-1$
$\mathrm{ln}\left(5\right)$
$\frac{5}{\mathrm{ln}\left(5\right)}-1$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=1$
$2$
$\frac{4}{3}$
$\frac{3}{4}$
$1$
Question 10:   Calculate the area of the figure defined by $y=2x$ , $x=2$ and $y=0$
$2$
$5$
$4$
$-4$