Calculus Examples

$\int 2 x^{2} (4 x - 1) d x$

Step 1

Multiply $2 x^{2} (4 x - 1)$ .

$\int 2 x^{2} (4 x) + 2 x^{2} \cdot - 1 d x$

Step 2

Simplify.

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Step 2.1

Multiply $4$ by $2$ .

$\int 8 x^{2} x + 2 x^{2} \cdot - 1 d x$

Step 2.2

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

$\int 8 (x^{1} x^{2}) + 2 x^{2} \cdot - 1 d x$

Step 2.3

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

$\int 8 x^{1 + 2} + 2 x^{2} \cdot - 1 d x$

Step 2.4

Add $1$ and $2$ .

$\int 8 x^{3} + 2 x^{2} \cdot - 1 d x$

Step 2.5

Multiply $- 1$ by $2$ .

$\int 8 x^{3} - 2 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 8 x^{3} d x + \int - 2 x^{2} d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{3} d x + \int - 2 x^{2} d x$

Step 5

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

$8 (\frac{1}{4} x^{4} + C) + \int - 2 x^{2} d x$

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$8 (\frac{1}{4} x^{4} + C) - 2 \int x^{2} d x$

Step 7

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

$8 (\frac{1}{4} x^{4} + C) - 2 (\frac{1}{3} x^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$8 (\frac{1}{4}) x^{4} - 2 (\frac{1}{3}) x^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $8$ and $\frac{1}{4}$ .

$\frac{8}{4} x^{4} - 2 (\frac{1}{3}) x^{3} + C$

Step 8.2.2

Cancel the common factor of $8$ and $4$ .

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Step 8.2.2.1

Factor $4$ out of $8$ .

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

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $4$ out of $4$ .

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

Step 8.2.2.2.2

Cancel the common factor.

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

Step 8.2.2.2.3

Rewrite the expression.

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

Step 8.2.2.2.4

Divide $2$ by $1$ .

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

Step 8.2.3

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 8.2.4

Move the negative in front of the fraction.

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