totalmente80sm9

2021-11-22

Evaluate the given integral.
$\int \frac{x}{{x}^{2}+1}dx$

Todd Williams

Step 1
Given integral, $\int \frac{x}{{x}^{2}+1}dx$
we have to evaluate the given integral.
Step 2
$\int \frac{x}{{x}^{2}+1}dx$
let ${x}^{2}+1=t⇒2xdx=dt⇒dx=\frac{dt}{2}$
$⇒\int \frac{x}{{x}^{2}+1}dx=\int \frac{1}{t}\frac{dt}{2}=\int \frac{dt}{2t}$
$=\frac{1}{2}\int \frac{dt}{t}$
we know $\int \frac{1}{x}dx=\mathrm{log}x+cons\mathrm{tan}tt$
$⇒\int \frac{x}{{x}^{2}+1}dx=\frac{1}{2}\mathrm{log}t+cons\mathrm{tan}tt$
substitue ${x}^{2}+1=t$
$⇒\int \frac{x}{{x}^{2}+1}dx=\frac{1}{2}\mathrm{log}\left({x}^{2}+1\right)+cons\mathrm{tan}tt$

Step 1: Use Integration by Substitution.
Let
Step 2: Using u and du above, rewrite $\int \frac{x}{{x}^{2}+1}dx$.
$\int \frac{1}{2u}du$
Step 3: Use Constant Factor Rule: $\int cf\left(x\right)dx=c\int f\left(x\right)dx$.
$\frac{1}{2}\int \frac{1}{u}du$
Step 4: The derivative of .
$\frac{\mathrm{ln}u}{2}$
Step 5: Substitute $u={x}^{2}+1$ back into the original integral.
$\frac{\mathrm{ln}\left({x}^{2}+1\right)}{2}$
$\frac{\mathrm{ln}\left({x}^{2}+1\right)}{2}+C$