Calculus Examples

$\int x^{4} - 3 x^{3} + 6 x^{2} - 7 d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{4} d x + \int - 3 x^{3} d x + \int 6 x^{2} d x + \int - 7 d x$

Step 2

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

$\frac{1}{5} x^{5} + C + \int - 3 x^{3} d x + \int 6 x^{2} d x + \int - 7 d x$

Step 3

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

$\frac{1}{5} x^{5} + C - 3 \int x^{3} d x + \int 6 x^{2} d x + \int - 7 d x$

Step 4

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

$\frac{1}{5} x^{5} + C - 3 (\frac{1}{4} x^{4} + C) + \int 6 x^{2} d x + \int - 7 d x$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{5} x^{5} + C - 3 (\frac{1}{4} x^{4} + C) + 6 \int x^{2} d x + \int - 7 d x$

Step 6

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

$\frac{1}{5} x^{5} + C - 3 (\frac{1}{4} x^{4} + C) + 6 (\frac{1}{3} x^{3} + C) + \int - 7 d x$

Step 7

Apply the constant rule.

$\frac{1}{5} x^{5} + C - 3 (\frac{1}{4} x^{4} + C) + 6 (\frac{1}{3} x^{3} + C) - 7 x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

$\frac{1}{5} x^{5} + C - 3 (\frac{x^{4}}{4} + C) + 6 (\frac{1}{3} x^{3} + C) - 7 x + C$

Step 8.1.2

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

$\frac{1}{5} x^{5} + C - 3 (\frac{x^{4}}{4} + C) + 6 (\frac{x^{3}}{3} + C) - 7 x + C$

Step 8.2

Simplify.

$\frac{x^{5}}{5} - \frac{3 x^{4}}{4} + 2 x^{3} - 7 x + C$

Step 9

Reorder terms.

$\frac{1}{5} x^{5} - \frac{3}{4} x^{4} + 2 x^{3} - 7 x + C$