Calculus Examples

$\int (x^{\frac{3}{2}} + 2 x + 1) d x$

Step 1

Remove parentheses.

$\int x^{\frac{3}{2}} + 2 x + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{\frac{3}{2}} d x + \int 2 x d x + \int d x$

Step 3

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + \int 2 x d x + \int d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{2}{5} x^{\frac{5}{2}} + C + 2 \int x d x + \int d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + 2 (\frac{1}{2} x^{2} + C) + \int d x$

Step 6

Apply the constant rule.

$\frac{2}{5} x^{\frac{5}{2}} + C + 2 (\frac{1}{2} x^{2} + C) + x + C$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + 2 (\frac{x^{2}}{2} + C) + x + C$

Step 7.2

Simplify.

$\frac{2 x^{\frac{5}{2}}}{5} + x^{2} + x + C$

Step 7.3

Reorder terms.

$\frac{2}{5} x^{\frac{5}{2}} + x^{2} + x + C$