Calculus

Antiderivatives

Given $\displaystyle{f(x) = \int \frac{2x + 49}{x^2+9x+14} dx}$ and $f(0) = 1,$ find the function $f(x).$

9\ln \lvert \frac{x+2}{2} \rvert + 7\ln \lvert \frac{7}{x+7} \rvert +1

7\ln \lvert \frac{x+2}{2} \rvert + 5\ln \lvert \frac{7}{x+7} \rvert +1

10\ln \lvert \frac{x+2}{2} \rvert + 8\ln \lvert \frac{7}{x+7} \rvert +1

8\ln \lvert \frac{x+2}{2} \rvert + 6\ln \lvert \frac{7}{x+7} \rvert +1

Evaluate $\int_{1}^{8} \frac{4}{x^3} dx .$

\frac{65}{32}

\frac{251}{128}

\frac{31}{8}

\frac{63}{32}

If $\displaystyle{f(x) = \int \frac{8x-9}{(x-3)(x-8)} dx},$ what is $f(5)-f(1)?$

11 \ln \frac{3}{7}

11 \ln \frac{1}{3}

11 \ln \frac{4}{7}

11 \ln \frac{3}{8}

If $\displaystyle{f(x)=\int \frac{38x-2}{19x^2-2x-2e}dx}$ and $f(0)=1+\ln 2-\ln 19,$ what is the value of $f(-e)?$

Evaluate the indefinite integral $\int \frac{x^2+17}{x-3}dx.$ (Use $C$ as the constant of integration.)

\frac{1}{2}x^2+26\ln \lvert{x-3}\rvert+C

\frac{1}{2}x^2-3x-26\ln \lvert{x-3}\rvert+C

x^2+3x+26\ln \lvert{x-3}\rvert+C

\frac{1}{2}x^2+3x+26\ln \lvert{x-3}\rvert+C