Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 4

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify.

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

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Combine $- 2$ and $\frac{1}{4}$ .

$\frac{1}{5} x^{5} + \frac{- 2}{4} x^{4} + C$

Step 5.2.2

Cancel the common factor of $- 2$ and $4$ .

Tap for more steps...

Step 5.2.2.1

Factor $2$ out of $- 2$ .

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

Step 5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.2.1

Factor $2$ out of $4$ .

$\frac{1}{5} x^{5} + \frac{2 \cdot - 1}{2 \cdot 2} x^{4} + C$

Step 5.2.2.2.2

Cancel the common factor.

$\frac{1}{5} x^{5} + \frac{2 \cdot - 1}{2 \cdot 2} x^{4} + C$

Step 5.2.2.2.3

Rewrite the expression.

$\frac{1}{5} x^{5} + \frac{- 1}{2} x^{4} + C$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{1}{5} x^{5} - \frac{1}{2} x^{4} + C$