Calculus Examples

$\int 8 x^{3} - 9 x^{2} + 4 d x$

Step 1

Split the single integral into multiple integrals.

$\int 8 x^{3} d x + \int - 9 x^{2} d x + \int 4 d x$

Step 2

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

$8 \int x^{3} d x + \int - 9 x^{2} d x + \int 4 d x$

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$8 (\frac{1}{4} x^{4} + C) + \int - 9 x^{2} d x + \int 4 d x$

Step 4

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

$8 (\frac{1}{4} x^{4} + C) - 9 \int x^{2} d x + \int 4 d x$

Step 5

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

$8 (\frac{1}{4} x^{4} + C) - 9 (\frac{1}{3} x^{3} + C) + \int 4 d x$

Step 6

Apply the constant rule.

$8 (\frac{1}{4} x^{4} + C) - 9 (\frac{1}{3} x^{3} + C) + 4 x + C$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$8 (\frac{x^{4}}{4} + C) - 9 (\frac{1}{3} x^{3} + C) + 4 x + C$

Step 7.1.2

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

$8 (\frac{x^{4}}{4} + C) - 9 (\frac{x^{3}}{3} + C) + 4 x + C$

Step 7.2

Simplify.

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