Calculus Examples

$\int (6 x^{6} + 3 x^{5} + 5 x - 4) d x$

Step 1

Remove parentheses.

$\int 6 x^{6} + 3 x^{5} + 5 x - 4 d x$

Step 2

Split the single integral into multiple integrals.

$\int 6 x^{6} d x + \int 3 x^{5} d x + \int 5 x d x + \int - 4 d x$

Step 3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x^{6} d x + \int 3 x^{5} d x + \int 5 x d x + \int - 4 d x$

Step 4

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$6 (\frac{1}{7} x^{7} + C) + \int 3 x^{5} d x + \int 5 x d x + \int - 4 d x$

Step 5

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

$6 (\frac{1}{7} x^{7} + C) + 3 \int x^{5} d x + \int 5 x d x + \int - 4 d x$

Step 6

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$6 (\frac{1}{7} x^{7} + C) + 3 (\frac{1}{6} x^{6} + C) + \int 5 x d x + \int - 4 d x$

Step 7

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$6 (\frac{1}{7} x^{7} + C) + 3 (\frac{1}{6} x^{6} + C) + 5 \int x d x + \int - 4 d x$

Step 8

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

$6 (\frac{1}{7} x^{7} + C) + 3 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{2} x^{2} + C) + \int - 4 d x$

Step 9

Apply the constant rule.

$6 (\frac{1}{7} x^{7} + C) + 3 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{2} x^{2} + C) - 4 x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$6 (\frac{x^{7}}{7} + C) + 3 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{2} x^{2} + C) - 4 x + C$

Step 10.1.2

Combine $\frac{1}{6}$ and $x^{6}$ .

$6 (\frac{x^{7}}{7} + C) + 3 (\frac{x^{6}}{6} + C) + 5 (\frac{1}{2} x^{2} + C) - 4 x + C$

Step 10.1.3

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

$6 (\frac{x^{7}}{7} + C) + 3 (\frac{x^{6}}{6} + C) + 5 (\frac{x^{2}}{2} + C) - 4 x + C$

Step 10.2

Simplify.

$\frac{6 x^{7}}{7} + \frac{x^{6}}{2} + \frac{5 x^{2}}{2} - 4 x + C$

Step 10.3

Reorder terms.

$\frac{6}{7} x^{7} + \frac{1}{2} x^{6} + \frac{5}{2} x^{2} - 4 x + C$