Calculus Examples

$\int - 9 x^{2} + 10 x d x$

Step 1

Split the single integral into multiple integrals.

$\int - 9 x^{2} d x + \int 10 x d x$

Step 2

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$- 9 \int x^{2} d x + \int 10 x d x$

Step 3

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

$- 9 (\frac{1}{3} x^{3} + C) + \int 10 x d x$

Step 4

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$- 9 (\frac{1}{3} x^{3} + C) + 10 \int x d x$

Step 5

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

$- 9 (\frac{1}{3} x^{3} + C) + 10 (\frac{1}{2} x^{2} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$- 9 (\frac{1}{3}) x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $- 9$ and $\frac{1}{3}$ .

$\frac{- 9}{3} x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.2

Cancel the common factor of $- 9$ and $3$ .

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Step 6.2.2.1

Factor $3$ out of $- 9$ .

$\frac{3 \cdot - 3}{3} x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 3}{3 (1)} x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot - 3}{3 \cdot 1} x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.2.2.3

Rewrite the expression.

$\frac{- 3}{1} x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.2.2.4

Divide $- 3$ by $1$ .

$- 3 x^{3} + 10 (\frac{1}{2}) x^{2} + C$

Step 6.2.3

Combine $10$ and $\frac{1}{2}$ .

$- 3 x^{3} + \frac{10}{2} x^{2} + C$

Step 6.2.4

Cancel the common factor of $10$ and $2$ .

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Step 6.2.4.1

Factor $2$ out of $10$ .

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

Step 6.2.4.2

Cancel the common factors.

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Step 6.2.4.2.1

Factor $2$ out of $2$ .

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

Step 6.2.4.2.2

Cancel the common factor.

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

Step 6.2.4.2.3

Rewrite the expression.

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

Step 6.2.4.2.4

Divide $5$ by $1$ .

$- 3 x^{3} + 5 x^{2} + C$