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Integration Techniques

Integration U-substitution - Given U

Using the substitution $\ln x=t,$ which of the following is equal to $\int_{e}^{e^{6}} \frac{3(\ln x)^2}{x} dx ?$

\int_{1}^{6} 3 t^2e^t dt

\int_{0}^{6} 3 t^2 dt

\int_{0}^{6} \frac{3 t^2}{e^t} dt

\int_{1}^{6} 3 t^2 dt

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by Brilliant Staff

Evaluate $\displaystyle{\int_0^{3}\frac{12x+6}{x^2+x+1}dx.}$

\frac{10}{39}

\frac{1}{13}

{6}(\ln{13}-\ln{3})

{6}\ln{13}

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by Brilliant Staff

Using the substitution $u=\sin x,$ evaluate $\displaystyle{\int_0^\frac{\pi}{2}\sin^{10} x\cos xdx.}$

\frac{\pi}{11}

\frac{\pi}{10}

\frac{1}{11}

\frac{1}{10}

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by Brilliant Staff

Evaluate $\displaystyle{\int_0^\frac{\pi}{32}3\tan8xdx.}$

\frac{3}{16}\ln3

-\frac{3}{16}\ln3

\frac{3}{16}\ln2

-\frac{3}{16}\ln2

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by Brilliant Staff

Using the substitution $u=e^{x^2},$ evaluate $\displaystyle{\int_0^{3}12x e^{x^2}dx.}$

{6}(e^{9}-1)

{12}e^{3}

{12}(e^{3}-1)

{6}e^{9}

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by Brilliant Staff

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