Calculus Examples

$\int_{1}^{2} 3 x^{5} - 2 x^{3} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} 3 x^{5} d x + \int_{1}^{2} - 2 x^{3} d x$

Step 2

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

$3 \int_{1}^{2} x^{5} d x + \int_{1}^{2} - 2 x^{3} d x$

Step 3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$3 (\frac{1}{6} x^{6}]_{1}^{2}) + \int_{1}^{2} - 2 x^{3} d x$

Step 4

Combine $\frac{1}{6}$ and $x^{6}$ .

$3 (\frac{x^{6}}{6}]_{1}^{2}) + \int_{1}^{2} - 2 x^{3} d x$

Step 5

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

$3 (\frac{x^{6}}{6}]_{1}^{2}) - 2 \int_{1}^{2} x^{3} d x$

Step 6

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

$3 (\frac{x^{6}}{6}]_{1}^{2}) - 2 (\frac{1}{4} x^{4}]_{1}^{2})$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$3 (\frac{x^{6}}{6}]_{1}^{2}) - 2 (\frac{x^{4}}{4}]_{1}^{2})$

Step 7.2

Substitute and simplify.

Tap for more steps...

Step 7.2.1

Evaluate $\frac{x^{6}}{6}$ at $2$ and at $1$ .

$3 ((\frac{2^{6}}{6}) - \frac{1^{6}}{6}) - 2 (\frac{x^{4}}{4}]_{1}^{2})$

Step 7.2.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $1$ .

$3 (\frac{2^{6}}{6} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3

Simplify.

Tap for more steps...

Step 7.2.3.1

Raise $2$ to the power of $6$ .

$3 (\frac{64}{6} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.2

Cancel the common factor of $64$ and $6$ .

Tap for more steps...

Step 7.2.3.2.1

Factor $2$ out of $64$ .

$3 (\frac{2 (32)}{6} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.2.2.1

Factor $2$ out of $6$ .

$3 (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.2.2.2

Cancel the common factor.

$3 (\frac{2 \cdot 32}{2 \cdot 3} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.2.2.3

Rewrite the expression.

$3 (\frac{32}{3} - \frac{1^{6}}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.3

One to any power is one.

$3 (\frac{32}{3} - \frac{1}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.4

To write $\frac{32}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 (\frac{32}{3} \cdot \frac{2}{2} - \frac{1}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 7.2.3.5.1

Multiply $\frac{32}{3}$ by $\frac{2}{2}$ .

$3 (\frac{32 \cdot 2}{3 \cdot 2} - \frac{1}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.5.2

Multiply $3$ by $2$ .

$3 (\frac{32 \cdot 2}{6} - \frac{1}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.6

Combine the numerators over the common denominator.

$3 \frac{32 \cdot 2 - 1}{6} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.7

Simplify the numerator.

Tap for more steps...

Step 7.2.3.7.1

Multiply $32$ by $2$ .

$3 \frac{64 - 1}{6} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.7.2

Subtract $1$ from $64$ .

$3 (\frac{63}{6}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.8

Cancel the common factor of $63$ and $6$ .

Tap for more steps...

Step 7.2.3.8.1

Factor $3$ out of $63$ .

$3 \frac{3 (21)}{6} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.8.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.8.2.1

Factor $3$ out of $6$ .

$3 \frac{3 \cdot 21}{3 \cdot 2} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.8.2.2

Cancel the common factor.

$3 \frac{3 \cdot 21}{3 \cdot 2} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.8.2.3

Rewrite the expression.

$3 (\frac{21}{2}) - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.9

Combine $3$ and $\frac{21}{2}$ .

$\frac{3 \cdot 21}{2} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.10

Multiply $3$ by $21$ .

$\frac{63}{2} - 2 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 7.2.3.11

Raise $2$ to the power of $4$ .

$\frac{63}{2} - 2 (\frac{16}{4} - \frac{1^{4}}{4})$

Step 7.2.3.12

Cancel the common factor of $16$ and $4$ .

Tap for more steps...

Step 7.2.3.12.1

Factor $4$ out of $16$ .

$\frac{63}{2} - 2 (\frac{4 \cdot 4}{4} - \frac{1^{4}}{4})$

Step 7.2.3.12.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.12.2.1

Factor $4$ out of $4$ .

$\frac{63}{2} - 2 (\frac{4 \cdot 4}{4 (1)} - \frac{1^{4}}{4})$

Step 7.2.3.12.2.2

Cancel the common factor.

$\frac{63}{2} - 2 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1^{4}}{4})$

Step 7.2.3.12.2.3

Rewrite the expression.

$\frac{63}{2} - 2 (\frac{4}{1} - \frac{1^{4}}{4})$

Step 7.2.3.12.2.4

Divide $4$ by $1$ .

$\frac{63}{2} - 2 (4 - \frac{1^{4}}{4})$

Step 7.2.3.13

One to any power is one.

$\frac{63}{2} - 2 (4 - \frac{1}{4})$

Step 7.2.3.14

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{63}{2} - 2 (4 \cdot \frac{4}{4} - \frac{1}{4})$

Step 7.2.3.15

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

$\frac{63}{2} - 2 (\frac{4 \cdot 4}{4} - \frac{1}{4})$

Step 7.2.3.16

Combine the numerators over the common denominator.

$\frac{63}{2} - 2 \frac{4 \cdot 4 - 1}{4}$

Step 7.2.3.17

Simplify the numerator.

Tap for more steps...

Step 7.2.3.17.1

Multiply $4$ by $4$ .

$\frac{63}{2} - 2 \frac{16 - 1}{4}$

Step 7.2.3.17.2

Subtract $1$ from $16$ .

$\frac{63}{2} - 2 (\frac{15}{4})$

Step 7.2.3.18

Combine $- 2$ and $\frac{15}{4}$ .

$\frac{63}{2} + \frac{- 2 \cdot 15}{4}$

Step 7.2.3.19

Multiply $- 2$ by $15$ .

$\frac{63}{2} + \frac{- 30}{4}$

Step 7.2.3.20

Cancel the common factor of $- 30$ and $4$ .

Tap for more steps...

Step 7.2.3.20.1

Factor $2$ out of $- 30$ .

$\frac{63}{2} + \frac{2 (- 15)}{4}$

Step 7.2.3.20.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.20.2.1

Factor $2$ out of $4$ .

$\frac{63}{2} + \frac{2 \cdot - 15}{2 \cdot 2}$

Step 7.2.3.20.2.2

Cancel the common factor.

$\frac{63}{2} + \frac{2 \cdot - 15}{2 \cdot 2}$

Step 7.2.3.20.2.3

Rewrite the expression.

$\frac{63}{2} + \frac{- 15}{2}$

Step 7.2.3.21

Move the negative in front of the fraction.

$\frac{63}{2} - \frac{15}{2}$

Step 7.2.3.22

Combine the numerators over the common denominator.

$\frac{63 - 15}{2}$

Step 7.2.3.23

Subtract $15$ from $63$ .

$\frac{48}{2}$

Step 7.2.3.24

Cancel the common factor of $48$ and $2$ .

Tap for more steps...

Step 7.2.3.24.1

Factor $2$ out of $48$ .

$\frac{2 \cdot 24}{2}$

Step 7.2.3.24.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.24.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 24}{2 (1)}$

Step 7.2.3.24.2.2

Cancel the common factor.

$\frac{2 \cdot 24}{2 \cdot 1}$

Step 7.2.3.24.2.3

Rewrite the expression.

$\frac{24}{1}$

Step 7.2.3.24.2.4

Divide $24$ by $1$ .

$24$

Step 8