Integration Techniques

Integration with Partial Fractions

If $f(x)=\int \frac{4x}{x^2-2x-3} dx$ and $f(0)=15 \ln 3,$ what is the value of $f(2)?$

13\ln 3

\ln 3

14\ln 3

12\ln 3

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by Brilliant Staff

If $f(x)$ is a function such that $f'(x)=\frac{1}{25x^2-1}$ and $f(0)=7,$ what is the value of $f(1)?$

-7+\frac{\ln 4+\ln 6}{10}

7+\frac{\ln 4-\ln 6}{10}

7+\frac{\ln 4-\ln 6}{5}

\frac{\ln 4-\ln 6}{5}

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by Brilliant Staff

What is the indefinite integral $\int \frac{x+2}{(x-4)(x-5)} dx?$ (Use $C$ for the constant of integration.)

\ln \lvert{x-4}\rvert+ \ln \lvert{x-5}\rvert+C

-6\ln \lvert {x-{4}} \rvert + 7 \ln \lvert{x-5}\rvert+C

\ln \lvert{x+2}\rvert-\ln \lvert{x-4}\rvert+ \ln \lvert{x-5}\rvert+C

-6\ln \lvert{x-10}\rvert+7 \ln \lvert{x-5}\rvert+C

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by Brilliant Staff

If $f(x)=\int \frac{12x^2}{(x-1)(x^2+x+1)} dx$ and $f(0)=0,$ what is the value of $\displaystyle{e^{\frac{f(9)}{4}}}?$

712 568 584 728

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by Brilliant Staff

If $a$ is a nonzero real number, what is $\int \frac{35}{x^2 - a^2} dx?$

Details and assumptions

Use $C$ as the constant of integration.

\frac{35}{2a} \ln \lvert \frac{x+a}{x-a} \rvert +C

\frac{35}{2a} \ln \lvert \frac{x-a}{x+35} \rvert +C

\frac{37}{2a} \ln \lvert \frac{x-a}{x+a} \rvert +C

\frac{35}{2a} \ln \lvert \frac{x-a}{x+a} \rvert +C

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by Brilliant Staff