Math / Calculus

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

Studdy Solution

STEP 1

1. The integral $\int \sqrt{x^{2}+1} x \, dx$ is a standard integral that can be solved using substitution.
2. The function inside the integral involves a composition of functions, suggesting the use of substitution.

STEP 2

1. Choose an appropriate substitution to simplify the integral.
2. Perform the substitution and simplify the integral.
3. Integrate the simplified expression.
4. Substitute back to the original variable.

STEP 3

Choose a substitution. Let $u = x^2 + 1$ . Then, compute the differential $du$ :
$u = x^2 + 1$ $\frac{du}{dx} = 2x$ $du = 2x \, dx$
To match the integral, solve for $x \, dx$ :
$x \, dx = \frac{1}{2} du$

STEP 4

Substitute $u$ and $du$ into the integral:
$\int \sqrt{x^2 + 1} \, x \, dx = \int \sqrt{u} \cdot \frac{1}{2} \, du$
Simplify the integral:
$= \frac{1}{2} \int u^{1/2} \, du$

STEP 5

Integrate the simplified expression:
$\frac{1}{2} \int u^{1/2} \, du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C$ $= \frac{1}{3} u^{3/2} + C$

STEP 6

Substitute back to the original variable $x$ :
$u = x^2 + 1$ $\frac{1}{3} u^{3/2} + C = \frac{1}{3} (x^2 + 1)^{3/2} + C$
The evaluated integral is:
$\boxed{\frac{1}{3} (x^2 + 1)^{3/2} + C}$

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