# Test: Integration III - Ambitious

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Question 1:   Find the antiderivative of the function $f\left(x\right)={\left(x-3\right)}^{2}$
$F\left(x\right)=\frac{{x}^{3}}{3}-3{x}^{2}+9x$
$F\left(x\right)=\frac{{x}^{3}}{3}-3{x}^{2}+9x+C$
$F\left(x\right)={x}^{3}-6{x}^{2}+9x+C$
$F\left(x\right)=\frac{{x}^{3}}{3}-3{x}^{2}+C$
Question 2:   Find the antiderivative of the function $f\left(x\right)={e}^{6x}$
$F\left(x\right)=6{e}^{6x}+C$
$F\left(x\right)=\frac{{e}^{x}}{6}+C$
$F\left(x\right)=\frac{{e}^{6x}}{6}$
$F\left(x\right)=\frac{{e}^{6x}}{6}+C$
Question 3:   Find the antiderivative of the function $f\left(x\right)=\mathrm{sin}\left(\frac{5x}{3}\right)$
$F\left(x\right)=-\mathrm{cos}\left(\frac{5x}{3}\right)+C$
$F\left(x\right)=-\frac{3}{5}\mathrm{cos}\left(\frac{5x}{3}\right)$
$F\left(x\right)=-\frac{3}{5}\mathrm{cos}\left(\frac{5x}{3}\right)+C$
$F\left(x\right)=\frac{3}{5}\mathrm{cos}\left(\frac{5x}{3}\right)+C$
Question 4:   Find the equation of integral for the area of the shaded region:

$S=\int \left(3x\right)dx$
$S=\int \left(3x-1\right)dx$
$S=\int \left(3x+1\right)dx$
$S=\int \left(3x-7x\right)dx$
Question 5:   Find the formula for the area of the shaded region:

$S=\int \left(7x+5\right)dx$
$S=\int \left(2\right)dx$
$S=\int \left(2x-1\right)dx$
$S=\int \left(2x\right)dx$
Question 6:   Find the integral of the function $f\left(x\right)=\frac{1}{7+{x}^{2}}$
$F\left(x\right)=\sqrt{7}\mathrm{arctan}\left(\frac{x}{\sqrt{7}}\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{7}}\mathrm{arctan}\left(x\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{7}}\mathrm{arctan}\left(\frac{x}{\sqrt{7}}\right)+C$
$F\left(x\right)=\frac{1}{\sqrt{7}}\mathrm{arctan}\left(\frac{x}{\sqrt{7}}\right)$
Question 7:   Calculate $\underset{1}{\overset{2}{\int }}\left(4{x}^{3}+2x\right)dx$
$18$
$16$
$20$
$12$
Question 8:   Calculate $\underset{0}{\overset{2}{\int }}\left({x}^{2}+1\right)dx$
$4$
$8$
$5$
$7$
Question 9:   Calculate the area of the figure defined by $y={x}^{2}$ and $y=9$
$36$
$9$
$\frac{1}{36}$
$2$
Question 10:   Calculate the area of the figure defined by $y=x$, $x=5$ and $y=0$
$7$
$12.5$
$14$
$13.5$