Definite Integration

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\log }_e\left(\sin 2x\right)dx=$

The value of ${\int }_0^1 \frac{x^2}{1+x^2}dx$ is

${\int }_0^{\infty }\left(a^{-x}-b^{-x}\right)dx=$

${\int }_0^{\frac{2}{3}} \frac{dx}{\sqrt{4-9x^2}}=$

Evaluate: ${\int }_2^{3}x\left(\log x^2\right)dx$

Evaluate: ${\int }_0^1\left(2-x\right)dx$

Find the value of the constant $a,$ if ${\int }_a^{2a}\left(x^2+ax+2\right)dx=\frac{73a}{2} \text{and} a>0.$

Evaluate the integral ${\int }_{-3}^{-2}x{\left(x+1\right)}^{2}dx$.

${\int }_0^{\mathrm{a}}\frac{\sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{a}-\mathrm{x}}}\mathrm{dx}$

Evaluate ${\int }_2^{3}x\left(\log x^2\right)dx$

${\int }_0^{\mathrm{\pi }/2}\frac{\log (1+{\mathrm{xsin}}^2\mathrm{\theta })}{{\sin }^2\mathrm{\theta }}\mathrm{d\theta },\mathrm{x}\ge 0$

Integrate ${\int }_0^2\left[{\left(x-1\right)}^2\tan \left(x-1\right)-{\left(x-1\right)}^3+1\right]dx$

Find $k:$

$3{\int }_2^k\left(2x^5-x^2\right)dx=-54$

${\int }_{-1}^1\mathrm{f}\left(\mathrm{x}\right) \mathrm{dx}$ where $\mathrm{f}\left(\mathrm{x}\right)=min\{\mathrm{x}+1,\sqrt{1-\mathrm{x}}\}$

${\int }_{-1}^2\left|{\mathrm{x}}^3-\mathrm{x}\right| \mathrm{dx}$

Evaluate ${\int }_0^1\frac{{\tan }^{-1}x}{1+x^2}dx$.

Evaluate the following integrals:

 ${\int }_0^{\frac{\pi }{4}}{\sin }^3 2t \cos 2t dt$

Evaluate the following integrals:

 ${\int }_1^2\frac{x dx}{\left(x+1\right)\left(x+2\right)}$

Evaluate the following integrals:

 ${\int }_4^9\frac{\sqrt{x}}{{\left(30-x^{\frac{3}{2}}\right)}^2}dx$

Evaluate ${\int }_0^2{e}^x dx$ as the limit of a sum.