totalmente80sm9

2021-11-22

Evaluate the given integral.

$\int \frac{x}{{x}^{2}+1}dx$

Todd Williams

Beginner2021-11-23Added 18 answers

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$x}^{2}+1=t\Rightarrow 2xdx=dt\Rightarrow dx=\frac{dt}{2$

$\Rightarrow \int \frac{x}{{x}^{2}+1}dx=\int \frac{1}{t}\frac{dt}{2}=\int \frac{dt}{2t}$

$=\frac{1}{2}\int \frac{dt}{t}$

we know$\int \frac{1}{x}dx=\mathrm{log}x+cons\mathrm{tan}tt$

$\Rightarrow \int \frac{x}{{x}^{2}+1}dx=\frac{1}{2}\mathrm{log}t+cons\mathrm{tan}tt$

substitue${x}^{2}+1=t$

$\Rightarrow \int \frac{x}{{x}^{2}+1}dx=\frac{1}{2}\mathrm{log}({x}^{2}+1)+cons\mathrm{tan}tt$

this is the required answer.

Given integral,

we have to evaluate the given integral.

Step 2

let

we know

substitue

this is the required answer.

Pulad1971

Beginner2021-11-24Added 22 answers

Step 1: Use Integration by Substitution.

Let$u={x}^{2}+1,du=2xdx,\text{}then\text{}xdx=\frac{1}{2}du$

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$\mathrm{ln}x\text{}is\text{}\frac{1}{x}$ .

$\frac{\mathrm{ln}u}{2}$

Step 5: Substitute$u={x}^{2}+1$ back into the original integral.

$\frac{\mathrm{ln}({x}^{2}+1)}{2}$

Step 6: Add constant.

$\frac{\mathrm{ln}({x}^{2}+1)}{2}+C$

Let

Step 2: Using u and du above, rewrite

Step 3: Use Constant Factor Rule:

Step 4: The derivative of

Step 5: Substitute

Step 6: Add constant.

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