Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor of $3$ .

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Rewrite the expression.

$\sin (x) + 1 x^{3} + C$

Step 6.2.3

Multiply $x^{3}$ by $1$ .

$\sin (x) + x^{3} + C$