Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

int(x)/(sqrt(x^(2)+1))dx

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

View step-by-step help

View step-by-step help

Find indefinite integrals using the substitution and by parts

Full solution

Q.

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

Simplify the integral: Step $1$ : Simplify the integral. $\newline$ We start by recognizing a substitution that simplifies the integral. Let $u = x^2 + 1$ , then $du = 2x dx$ . This means $x dx = \frac{du}{2}$ .
Substitute and adjust: Step $2$ : Substitute and adjust the integral. $\newline$ Substitute $u$ and $du$ into the integral: $\newline$ $\int\frac{x}{\sqrt{x^2 + 1}} dx = \int\frac{x}{\sqrt{u}} \cdot \left(\frac{du}{2}\right)$ $\newline$ $= \frac{1}{2} \int\frac{1}{\sqrt{u}} \cdot du$
Integrate with new variable: Step $3$ : Integrate using the new variable. $\newline$ The integral of $\frac{1}{\sqrt{u}}$ with respect to $u$ is $2\sqrt{u}$ . So, $\newline$ $\frac{1}{2} \int \frac{1}{\sqrt{u}} \, du = \frac{1}{2} \cdot 2 \cdot \sqrt{u} + C$ $\newline$ $= \sqrt{u} + C$
Substitute back to x: Step $4$ : Substitute back to x. $\newline$ Since $u = x^2 + 1$ , $\sqrt{u} = \sqrt{x^2 + 1}$ . Therefore, $\newline$ $\sqrt{u} + C = \sqrt{x^2 + 1} + C$

More problems from Find indefinite integrals using the substitution and by parts

Question

Evaluate the integral.

\newline

\int-x 4^{-3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{3} e^{-2 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int x 5^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-2 x e^{2 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-2 x \cos (-2 x) d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-x \cos (-3 x) d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 5^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-6 x 4^{-4 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 2^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-3 x \sin (-2 x) d x

\newline

Posted 2 months ago