Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

$\int 2 \sqrt{x} d x + \int - 3 x^{2} d x + \int d x$

Step 2

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

$2 \int \sqrt{x} d x + \int - 3 x^{2} d x + \int d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

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

Step 5

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 7

Apply the constant rule.

$2 (\frac{2}{3} x^{\frac{3}{2}} + C) - 3 (\frac{1}{3} x^{3} + C) + x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$2 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 3 (\frac{1}{3} x^{3} + C) + x + C$

Step 8.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 3 (\frac{x^{3}}{3} + C) + x + C$

Step 8.2

Simplify.

$\frac{4 x^{\frac{3}{2}}}{3} - x^{3} + x + C$

Step 8.3

Reorder terms.

$\frac{4}{3} x^{\frac{3}{2}} - x^{3} + x + C$