Calculus Examples

$\int 9 x^{2} - 8 x + 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int 9 x^{2} d x + \int - 8 x d x + \int 3 d x$

Step 2

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x^{2} d x + \int - 8 x d x + \int 3 d x$

Step 3

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

$9 (\frac{1}{3} x^{3} + C) + \int - 8 x d x + \int 3 d x$

Step 4

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$9 (\frac{1}{3} x^{3} + C) - 8 \int x d x + \int 3 d x$

Step 5

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

$9 (\frac{1}{3} x^{3} + C) - 8 (\frac{1}{2} x^{2} + C) + \int 3 d x$

Step 6

Apply the constant rule.

$9 (\frac{1}{3} x^{3} + C) - 8 (\frac{1}{2} x^{2} + C) + 3 x + C$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

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

$9 (\frac{x^{3}}{3} + C) - 8 (\frac{1}{2} x^{2} + C) + 3 x + C$

Step 7.1.2

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

$9 (\frac{x^{3}}{3} + C) - 8 (\frac{x^{2}}{2} + C) + 3 x + C$

Step 7.2

Simplify.

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