Calculus Examples

$\int 6 x^{2} - x + 7 d x$

Step 1

Split the single integral into multiple integrals.

$\int 6 x^{2} d x + \int - x d x + \int 7 d x$

Step 2

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

$6 \int x^{2} d x + \int - x d x + \int 7 d x$

Step 3

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

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

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Simplify.

$2 x^{3} - \frac{x^{2}}{2} + 7 x + C$

Step 8

Reorder terms.

$2 x^{3} - \frac{1}{2} x^{2} + 7 x + C$

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