evaluate the integral using integration by parts.\n int 2x^{3}e^{5x}mathrm{d}x \n int 2x^{3}e^{5x}mathrm{d}x=…

Explanation:

Step1: Recall integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = 2x^{3}$ and $\mathrm{d}v=e^{5x}\mathrm{d}x$. Then $\mathrm{d}u = 6x^{2}\mathrm{d}x$ and $v=\frac{1}{5}e^{5x}$. So, $\int 2x^{3}e^{5x}\mathrm{d}x=\frac{2}{5}x^{3}e^{5x}-\int\frac{6}{5}x^{2}e^{5x}\mathrm{d}x$.

Step2: Apply integration - by - parts again

For $\int\frac{6}{5}x^{2}e^{5x}\mathrm{d}x$, let $u=\frac{6}{5}x^{2}$ and $\mathrm{d}v = e^{5x}\mathrm{d}x$. Then $\mathrm{d}u=\frac{12}{5}x\mathrm{d}x$ and $v=\frac{1}{5}e^{5x}$. $\int\frac{6}{5}x^{2}e^{5x}\mathrm{d}x=\frac{6}{25}x^{2}e^{5x}-\int\frac{12}{25}xe^{5x}\mathrm{d}x$.

Step3: Apply integration - by - parts for the third time

For $\int\frac{12}{25}xe^{5x}\mathrm{d}x$, let $u=\frac{12}{25}x$ and $\mathrm{d}v = e^{5x}\mathrm{d}x$. Then $\mathrm{d}u=\frac{12}{25}\mathrm{d}x$ and $v=\frac{1}{5}e^{5x}$. $\int\frac{12}{25}xe^{5x}\mathrm{d}x=\frac{12}{125}xe^{5x}-\int\frac{12}{125}e^{5x}\mathrm{d}x$.

Step4: Integrate the remaining integral

$\int\frac{12}{125}e^{5x}\mathrm{d}x=\frac{12}{625}e^{5x}+C$.

Step5: Back - substitute

$\int 2x^{3}e^{5x}\mathrm{d}x=\frac{2}{5}x^{3}e^{5x}-\left(\frac{6}{25}x^{2}e^{5x}-\left(\frac{12}{25}xe^{5x}-\frac{12}{125}e^{5x}\right)\right)+C$ $=\frac{2}{5}x^{3}e^{5x}-\frac{6}{25}x^{2}e^{5x}+\frac{12}{125}xe^{5x}-\frac{12}{625}e^{5x}+C$ $=\frac{1}{625}e^{5x}(250x^{3}-150x^{2} + 60x-12)+C$

Answer:

$\frac{1}{625}e^{5x}(250x^{3}-150x^{2}+60x - 12)+C$