Calculus Examples

$\int (a x^{2} + b x + c) d x$

Step 1

Remove parentheses.

$\int a x^{2} + b x + c d x$

Step 2

Split the single integral into multiple integrals.

$\int a x^{2} d x + \int b x d x + \int c d x$

Step 3

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

$a \int x^{2} d x + \int b x d x + \int c d x$

Step 4

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

$a (\frac{1}{3} x^{3} + C) + \int b x d x + \int c d x$

Step 5

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

$a (\frac{1}{3} x^{3} + C) + b \int x d x + \int c d x$

Step 6

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

$a (\frac{1}{3} x^{3} + C) + b (\frac{1}{2} x^{2} + C) + \int c d x$

Step 7

Apply the constant rule.

$a (\frac{1}{3} x^{3} + C) + b (\frac{1}{2} x^{2} + C) + c x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

$a (\frac{x^{3}}{3} + C) + b (\frac{1}{2} x^{2} + C) + c x + C$

Step 8.1.2

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

$a (\frac{x^{3}}{3} + C) + b (\frac{x^{2}}{2} + C) + c x + C$

Step 8.2

Simplify.

$\frac{1}{3} a x^{3} + \frac{1}{2} b x^{2} + c x + C$