Calculus Examples

$\int_{2}^{4} 10 + 4 x - 3 x^{3} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{4} 10 d x + \int_{2}^{4} 4 x d x + \int_{2}^{4} - 3 x^{3} d x$

Step 2

Apply the constant rule.

$10 x]_{2}^{4} + \int_{2}^{4} 4 x d x + \int_{2}^{4} - 3 x^{3} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$10 x]_{2}^{4} + 4 \int_{2}^{4} x d x + \int_{2}^{4} - 3 x^{3} d x$

Step 4

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

$10 x]_{2}^{4} + 4 (\frac{1}{2} x^{2}]_{2}^{4}) + \int_{2}^{4} - 3 x^{3} d x$

Step 5

Combine $\frac{1}{2}$ and $x^{2}$ .

$10 x]_{2}^{4} + 4 (\frac{x^{2}}{2}]_{2}^{4}) + \int_{2}^{4} - 3 x^{3} d x$

Step 6

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

$10 x]_{2}^{4} + 4 (\frac{x^{2}}{2}]_{2}^{4}) - 3 \int_{2}^{4} x^{3} d x$

Step 7

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

$10 x]_{2}^{4} + 4 (\frac{x^{2}}{2}]_{2}^{4}) - 3 (\frac{1}{4} x^{4}]_{2}^{4})$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

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

$10 x]_{2}^{4} + 4 (\frac{x^{2}}{2}]_{2}^{4}) - 3 (\frac{x^{4}}{4}]_{2}^{4})$

Step 8.2

Substitute and simplify.

Tap for more steps...

Step 8.2.1

Evaluate $10 x$ at $4$ and at $2$ .

$(10 \cdot 4) - 10 \cdot 2 + 4 (\frac{x^{2}}{2}]_{2}^{4}) - 3 (\frac{x^{4}}{4}]_{2}^{4})$

Step 8.2.2

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

$10 \cdot 4 - 10 \cdot 2 + 4 (\frac{4^{2}}{2} - \frac{2^{2}}{2}) - 3 (\frac{x^{4}}{4}]_{2}^{4})$

Step 8.2.3

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

$10 \cdot 4 - 10 \cdot 2 + 4 (\frac{4^{2}}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4

Simplify.

Tap for more steps...

Step 8.2.4.1

Multiply $10$ by $4$ .

$40 - 10 \cdot 2 + 4 (\frac{4^{2}}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.2

Multiply $- 10$ by $2$ .

$40 - 20 + 4 (\frac{4^{2}}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.3

Subtract $20$ from $40$ .

$20 + 4 (\frac{4^{2}}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.4

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

$20 + 4 (\frac{16}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.5

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 8.2.4.5.1

Factor $2$ out of $16$ .

$20 + 4 (\frac{2 \cdot 8}{2} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.5.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.5.2.1

Factor $2$ out of $2$ .

$20 + 4 (\frac{2 \cdot 8}{2 (1)} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.5.2.2

Cancel the common factor.

$20 + 4 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.5.2.3

Rewrite the expression.

$20 + 4 (\frac{8}{1} - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.5.2.4

Divide $8$ by $1$ .

$20 + 4 (8 - \frac{2^{2}}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.6

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

$20 + 4 (8 - \frac{4}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.7

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 8.2.4.7.1

Factor $2$ out of $4$ .

$20 + 4 (8 - \frac{2 \cdot 2}{2}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.7.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.7.2.1

Factor $2$ out of $2$ .

$20 + 4 (8 - \frac{2 \cdot 2}{2 (1)}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.7.2.2

Cancel the common factor.

$20 + 4 (8 - \frac{2 \cdot 2}{2 \cdot 1}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.7.2.3

Rewrite the expression.

$20 + 4 (8 - \frac{2}{1}) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.7.2.4

Divide $2$ by $1$ .

$20 + 4 (8 - 1 \cdot 2) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.8

Multiply $- 1$ by $2$ .

$20 + 4 (8 - 2) - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.9

Subtract $2$ from $8$ .

$20 + 4 \cdot 6 - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.10

Multiply $4$ by $6$ .

$20 + 24 - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.11

Add $20$ and $24$ .

$44 - 3 ((\frac{4^{4}}{4}) - \frac{2^{4}}{4})$

Step 8.2.4.12

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

$44 - 3 (\frac{256}{4} - \frac{2^{4}}{4})$

Step 8.2.4.13

Cancel the common factor of $256$ and $4$ .

Tap for more steps...

Step 8.2.4.13.1

Factor $4$ out of $256$ .

$44 - 3 (\frac{4 \cdot 64}{4} - \frac{2^{4}}{4})$

Step 8.2.4.13.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.13.2.1

Factor $4$ out of $4$ .

$44 - 3 (\frac{4 \cdot 64}{4 (1)} - \frac{2^{4}}{4})$

Step 8.2.4.13.2.2

Cancel the common factor.

$44 - 3 (\frac{4 \cdot 64}{4 \cdot 1} - \frac{2^{4}}{4})$

Step 8.2.4.13.2.3

Rewrite the expression.

$44 - 3 (\frac{64}{1} - \frac{2^{4}}{4})$

Step 8.2.4.13.2.4

Divide $64$ by $1$ .

$44 - 3 (64 - \frac{2^{4}}{4})$

Step 8.2.4.14

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

$44 - 3 (64 - \frac{16}{4})$

Step 8.2.4.15

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

Tap for more steps...

Step 8.2.4.15.1

Factor $4$ out of $16$ .

$44 - 3 (64 - \frac{4 \cdot 4}{4})$

Step 8.2.4.15.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.15.2.1

Factor $4$ out of $4$ .

$44 - 3 (64 - \frac{4 \cdot 4}{4 (1)})$

Step 8.2.4.15.2.2

Cancel the common factor.

$44 - 3 (64 - \frac{4 \cdot 4}{4 \cdot 1})$

Step 8.2.4.15.2.3

Rewrite the expression.

$44 - 3 (64 - \frac{4}{1})$

Step 8.2.4.15.2.4

Divide $4$ by $1$ .

$44 - 3 (64 - 1 \cdot 4)$

Step 8.2.4.16

Multiply $- 1$ by $4$ .

$44 - 3 (64 - 4)$

Step 8.2.4.17

Subtract $4$ from $64$ .

$44 - 3 \cdot 60$

Step 8.2.4.18

Multiply $- 3$ by $60$ .

$44 - 180$

Step 8.2.4.19

Subtract $180$ from $44$ .

$- 136$

Step 9