"Evaluate the given integral. ∫xx2+1dx" solved and explained

totalmente80sm9

Answered question

2021-11-22

Evaluate the given integral.

\int \frac{x}{x^{2} + 1} d x

Todd Williams

Beginner2021-11-23Added 18 answers

Step 1
Given integral,

\int \frac{x}{x^{2} + 1} d x

we have to evaluate the given integral.
Step 2

\int \frac{x}{x^{2} + 1} d x

let

x^{2} + 1 = t \Rightarrow 2 x d x = d t \Rightarrow d x = \frac{d t}{2}

\Rightarrow \int \frac{x}{x^{2} + 1} d x = \int \frac{1}{t} \frac{d t}{2} = \int \frac{d t}{2 t}

= \frac{1}{2} \int \frac{d t}{t}

we know

\int \frac{1}{x} d x = \log x + c o n s \tan t t

\Rightarrow \int \frac{x}{x^{2} + 1} d x = \frac{1}{2} \log t + c o n s \tan t t

substitue

x^{2} + 1 = t

\Rightarrow \int \frac{x}{x^{2} + 1} d x = \frac{1}{2} \log (x^{2} + 1) + c o n s \tan t t

this is the required answer.

Pulad1971

Beginner2021-11-24Added 22 answers

Step 1: Use Integration by Substitution.
Let

u = x^{2} + 1, d u = 2 x d x, t h e n x d x = \frac{1}{2} d u

Step 2: Using u and du above, rewrite

\int \frac{x}{x^{2} + 1} d x

\int \frac{1}{2 u} d u

Step 3: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\frac{1}{2} \int \frac{1}{u} d u

Step 4: The derivative of

\ln x i s \frac{1}{x}

\frac{\ln u}{2}

Step 5: Substitute

u = x^{2} + 1

back into the original integral.

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

Step 6: Add constant.

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

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?