Calculus Examples

$\int (5 x^{4} - 4 x^{3} + 3 x^{2}) d x$

Step 1

Remove parentheses.

$\int 5 x^{4} - 4 x^{3} + 3 x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 x^{4} d x + \int - 4 x^{3} d x + \int 3 x^{2} d x$

Step 3

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

$5 \int x^{4} d x + \int - 4 x^{3} d x + \int 3 x^{2} d x$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

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

Step 5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

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

$x^{5} - x^{4} + 3 \frac{x^{3}}{3} + C$

Step 9.2.2

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

$x^{5} - x^{4} + \frac{3 x^{3}}{3} + C$

Step 9.2.3

Cancel the common factor of $3$ .

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Step 9.2.3.1

Cancel the common factor.

$x^{5} - x^{4} + \frac{3 x^{3}}{3} + C$

Step 9.2.3.2

Divide $x^{3}$ by $1$ .

$x^{5} - x^{4} + x^{3} + C$