Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Simplify.

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

Step 8.3

Simplify.

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Step 8.3.1

Combine $6$ and $\frac{x^{4}}{4}$ .

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

Step 8.3.2

Cancel the common factor of $6$ and $4$ .

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Step 8.3.2.1

Factor $2$ out of $6 x^{4}$ .

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

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $2$ out of $4$ .

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 8.4

Reorder terms.

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