evaluate the following integral using integration by parts.\n int se^{-21s} ds \n int se^{-21s} ds=square

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = s$ and $\mathrm{d}v=e^{-21s}\mathrm{d}s$. Then $\mathrm{d}u=\mathrm{d}s$ and $v=-\frac{1}{21}e^{-21s}$.

Step2: Substitute into the formula

$\int se^{-21s}\mathrm{d}s=u v-\int v\mathrm{d}u=s\left(-\frac{1}{21}e^{-21s}\right)-\int\left(-\frac{1}{21}e^{-21s}\right)\mathrm{d}s$.

Step3: Evaluate the remaining integral

$\int\left(-\frac{1}{21}e^{-21s}\right)\mathrm{d}s=\frac{1}{441}e^{-21s}+C$. So, $\int se^{-21s}\mathrm{d}s=-\frac{s}{21}e^{-21s}+\frac{1}{441}e^{-21s}+C$.

Answer:

$-\frac{s}{21}e^{-21s}+\frac{1}{441}e^{-21s}+C$