Calculus Examples

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

Step 1

Move $2$ to the left of $5 x^{2} - 3$ .

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

Step 2

Multiply $2 (5 x^{2} - 3) x$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $5$ by $2$ .

$\int 10 x^{2} x + 2 \cdot - 3 x d x$

Step 3.2

Raise $x$ to the power of $1$ .

$\int 10 (x^{1} x^{2}) + 2 \cdot - 3 x d x$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 10 x^{1 + 2} + 2 \cdot - 3 x d x$

Step 3.4

Add $1$ and $2$ .

$\int 10 x^{3} + 2 \cdot - 3 x d x$

Step 3.5

Multiply $2$ by $- 3$ .

$\int 10 x^{3} - 6 x d x$

Step 4

Split the single integral into multiple integrals.

$\int 10 x^{3} d x + \int - 6 x d x$

Step 5

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

$10 \int x^{3} d x + \int - 6 x d x$

Step 6

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

$10 (\frac{1}{4} x^{4} + C) + \int - 6 x d x$

Step 7

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

$10 (\frac{1}{4} x^{4} + C) - 6 \int x d x$

Step 8

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

$10 (\frac{1}{4} x^{4} + C) - 6 (\frac{1}{2} x^{2} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$10 (\frac{1}{4}) x^{4} - 6 (\frac{1}{2}) x^{2} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $10$ and $\frac{1}{4}$ .

$\frac{10}{4} x^{4} - 6 (\frac{1}{2}) x^{2} + C$

Step 9.2.2

Cancel the common factor of $10$ and $4$ .

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Step 9.2.2.1

Factor $2$ out of $10$ .

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

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2.2.2

Cancel the common factor.

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

Step 9.2.2.2.3

Rewrite the expression.

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

Step 9.2.3

Combine $- 6$ and $\frac{1}{2}$ .

$\frac{5}{2} x^{4} + \frac{- 6}{2} x^{2} + C$

Step 9.2.4

Cancel the common factor of $- 6$ and $2$ .

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Step 9.2.4.1

Factor $2$ out of $- 6$ .

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

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $2$ out of $2$ .

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

Step 9.2.4.2.2

Cancel the common factor.

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

Step 9.2.4.2.3

Rewrite the expression.

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

Step 9.2.4.2.4

Divide $- 3$ by $1$ .

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