Calculus Examples

$\int (x^{2} + 2 + 4 x^{4}) d x$

Step 1

Remove parentheses.

$\int x^{2} + 2 + 4 x^{4} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{2} d x + \int 2 d x + \int 4 x^{4} d x$

Step 3

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

$\frac{1}{3} x^{3} + C + \int 2 d x + \int 4 x^{4} d x$

Step 4

Apply the constant rule.

$\frac{1}{3} x^{3} + C + 2 x + C + \int 4 x^{4} d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{3} x^{3} + C + 2 x + C + 4 \int x^{4} d x$

Step 6

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{3} x^{3} + C + 2 x + C + 4 (\frac{1}{5} x^{5} + C)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$\frac{1}{3} x^{3} + C + 2 x + C + 4 (\frac{x^{5}}{5} + C)$

Step 7.2

Simplify.

$\frac{1}{3} x^{3} + 2 x + 4 (\frac{1}{5}) x^{5} + C$

Step 7.3

Combine $4$ and $\frac{1}{5}$ .

$\frac{1}{3} x^{3} + 2 x + \frac{4}{5} x^{5} + C$