eozoischgc

2021-12-27

Evaluate the following integrals.
$\int x{10}^{x}dx$

Jillian Edgerton

Step 1
Given: $I=\int x{\left(10\right)}^{x}dx$
for evaluating given integral, we will use integral by parts theorem
according to this theorem
$\int {f}^{\prime }\left(x\right)g\left(x\right)dx=g\left(x\right)\int {f}^{\prime }\left(x\right)dx-\int \left[\left({g}^{\prime }\left(x\right)\right)\int {f}^{\prime }\left(x\right)dx\right]dx+c$
Step 2
so,
$I=\int \left(x\right)\left({10}^{x}\right)dx$
$=x\int \left({10}^{x}\right)dx-\int \left[1\int {10}^{x}dx\right]dx$
$\left(\because \int {a}^{x}dx=\frac{{a}^{x}}{\mathrm{ln}a}+c\right)$
$=x\left(\frac{{10}^{x}}{\mathrm{ln}10}\right)-\int \left(\frac{{10}^{x}}{\mathrm{ln}10}\right)dx+c$
$=\frac{x\left({10}^{x}\right)}{\mathrm{ln}10}-\frac{1}{\mathrm{ln}10}\left(\frac{{10}^{x}}{\mathrm{ln}10}\right)+c$
hence, given integral is $\frac{x\left({10}^{x}\right)}{\mathrm{ln}10}-\frac{{10}^{x}}{{\left(\mathrm{ln}10\right)}^{2}}+c$.

Andrew Reyes

$\int x\cdot {10}^{x}dx$
Integration piece by piece: $\int f{g}^{\prime }=fg-\int {f}^{\prime }g$
$f=x,{g}^{\prime }={10}^{x}$
$=\frac{x\cdot {10}^{x}}{\mathrm{ln}\left(10\right)}-\int \frac{{10}^{x}}{\mathrm{ln}\left(10\right)}dx$
$\int \frac{{10}^{x}}{\mathrm{ln}\left(10\right)}dx$
Let's apply linearity:
$=\frac{1}{\mathrm{ln}\left(10\right)}\int {10}^{x}dx$
$\int {10}^{x}dx$
Integral of exponential function:
$\int {a}^{x}dx=\frac{{a}^{x}}{\mathrm{ln}\left(a\right)}$ at a=10:
$=\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}$
$\frac{1}{\mathrm{ln}\left(10\right)}\int {10}^{x}dx$
$=\frac{{10}^{x}}{{\mathrm{ln}}^{2}\left(10\right)}$
$\frac{x\cdot {10}^{x}}{\mathrm{ln}\left(10\right)}-\int \frac{{10}^{x}}{\mathrm{ln}\left(10\right)}dx$
$=\frac{x\cdot {10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{{10}^{x}}{{\mathrm{ln}}^{2}\left(10\right)}$
$\int x{10}^{x}dx$
$=\frac{x\cdot {10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{{10}^{x}}{{\mathrm{ln}}^{2}\left(10\right)}+C$
Let's rewrite / simplify:
$=\frac{\left(\mathrm{ln}\left(10\right)x-1\right)\cdot {10}^{x}}{{\mathrm{ln}}^{2}\left(10\right)}+C$

Vasquez

$\int x×{10}^{x}dx$
Prepare for integration by parts
u=x
$dv={10}^{x}dx$
du=dx
$v=\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}$
$x×\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}-\int \frac{{10}^{x}}{\mathrm{ln}\left(10\right)}dx$
$x×\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{1}{\mathrm{ln}\left(10\right)}×\int {10}^{x}dx$
$x×\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{1}{\mathrm{ln}\left(10\right)}×\frac{{10}^{x}}{\mathrm{ln}\left(10\right)}$
Simplify
$\frac{x×{10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{{10}^{x}}{\mathrm{ln}\left(10{\right)}^{2}}$
$\frac{x×{10}^{x}}{\mathrm{ln}\left(10\right)}-\frac{{10}^{x}}{\mathrm{ln}\left(10{\right)}^{2}}+C$