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Integration Techniques

Integration by Parts

Find the indefinite integral $\displaystyle{ 43\int{x \ln x dx} },$ using $C$ as the constant of integration.

\frac{43}{2} x^3\ln x - \frac{43}{4} x^2 + C

\frac{43}{2} x^2\ln x - \frac{43}{4} x^2 + C

\frac{43}{2} x^3\ln x - \frac{43}{4} x + C

\frac{43}{2} x^2\ln x - \frac{43}{4} x + C

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by Brilliant Staff

Evaluate the integral $\displaystyle{ 128\int{x \cos 3x dx} },$ using $C$ as the constant of integration.

\frac{128}{3}x^2\sin 3x + \frac{128}{9} x\cos 3x + C

\frac{128}{3}x^3\sin 3x + \frac{128}{9}x^2 \cos 3x + C

\frac{128}{3}\sin 3x + \frac{128}{9} x^2\cos 3x + C

\frac{128}{3}x\sin 3x + \frac{128}{9} \cos 3x + C

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Find the indefinite integral $\displaystyle{ \int{x \sqrt{x+19} dx} },$ using $C$ as the constant of integration.

\frac{2}{3}x(x+19)^{\frac{3}{2}} - \frac{4}{15}(x+19)^{\frac{5}{2}} + C

\frac{2}{3}x(x+19)^{\frac{5}{2}} - \frac{2}{15}(x+19)^{\frac{5}{2}} + C

\frac{1}{3}x(x+19)^{\frac{3}{2}} - \frac{7}{15}(x+19)^{\frac{5}{2}} + C

\frac{2}{3}x(x+19)^{\frac{1}{2}} - \frac{4}{15}(x+19)^{\frac{5}{2}} + C

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by Brilliant Staff

Evaluate the integral $\displaystyle{ 67\int{t^2 e^{3t} dt} },$ using $C$ as the constant of integration.

\frac{67}{3}t^2e^{3t} - \frac{134}{9} te^{3t} + \frac{134}{27} e^{3t} + C

\frac{67}{9} te^{3t} + \frac{134}{27} e^{3t} + C

\frac{67}{3}t^2e^{3t} - \frac{134}{9} te^{3t} + + C

\frac{67}{3}t^3e^{3t} - \frac{134}{9} t^2e^{3t} + \frac{134}{27} e^{3t} + C

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Evaluate the integral $\displaystyle{ 43\int _{ 0 }^{ 1 }{ \arctan x dx } },$ using $C$ as the constant of integration.

\frac{43}{2}\pi - 43\ln(\sqrt{5})

\frac{43}{4}\pi - 43\ln(\sqrt{2})

\frac{43}{6}\pi - 43\ln(\sqrt{3})

\frac{43}{2}\pi - 43\ln(\sqrt6)

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by Brilliant Staff