Calculus Examples

$\int (3 x^{3} + 5 x^{- 2} + 6 x^{2}) d x$

Step 1

Remove parentheses.

$\int 3 x^{3} + 5 x^{- 2} + 6 x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 x^{3} d x + \int 5 x^{- 2} d x + \int 6 x^{2} d x$

Step 3

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

$3 \int x^{3} d x + \int 5 x^{- 2} d x + \int 6 x^{2} d x$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{3 x^{4}}{4} - \frac{5}{x} + 6 \frac{x^{3}}{3} + C$

Step 9.2.2

Combine $6$ and $\frac{x^{3}}{3}$ .

$\frac{3 x^{4}}{4} - \frac{5}{x} + \frac{6 x^{3}}{3} + C$

Step 9.2.3

Cancel the common factor of $6$ and $3$ .

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Step 9.2.3.1

Factor $3$ out of $6 x^{3}$ .

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

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.2.2

Cancel the common factor.

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

Step 9.2.3.2.3

Rewrite the expression.

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

Step 9.2.3.2.4

Divide $2 x^{3}$ by $1$ .

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

Step 9.3

Reorder terms.

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