Calculus Examples

$\int_{- 2}^{2} (x^{7} + k) d x = 16$

Step 1

Remove parentheses.

$\int_{- 2}^{2} x^{7} + k d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{2} x^{7} d x + \int_{- 2}^{2} k d x$

Step 3

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{8} x^{8}]_{- 2}^{2} + \int_{- 2}^{2} k d x$

Step 4

Apply the constant rule.

$\frac{1}{8} x^{8}]_{- 2}^{2} + k x]_{- 2}^{2}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Combine $\frac{1}{8} x^{8}]_{- 2}^{2}$ and $k x]_{- 2}^{2}$ .

$\frac{1}{8} x^{8} + k x]_{- 2}^{2}$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $\frac{1}{8} x^{8} + k x$ at $2$ and at $- 2$ .

$(\frac{1}{8} \cdot 2^{8} + k \cdot 2) - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2

Simplify.

Tap for more steps...

Step 5.2.2.1

Raise $2$ to the power of $8$ .

$\frac{1}{8} \cdot 256 + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.2

Combine $\frac{1}{8}$ and $256$ .

$\frac{256}{8} + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.3

Cancel the common factor of $256$ and $8$ .

Tap for more steps...

Step 5.2.2.3.1

Factor $8$ out of $256$ .

$\frac{8 \cdot 32}{8} + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.3.2.1

Factor $8$ out of $8$ .

$\frac{8 \cdot 32}{8 (1)} + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.3.2.2

Cancel the common factor.

$\frac{8 \cdot 32}{8 \cdot 1} + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.3.2.3

Rewrite the expression.

$\frac{32}{1} + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.3.2.4

Divide $32$ by $1$ .

$32 + k \cdot 2 - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.4

Move $2$ to the left of $k$ .

$32 + 2 \cdot k - (\frac{1}{8} {(- 2)}^{8} + k \cdot - 2)$

Step 5.2.2.5

Raise $- 2$ to the power of $8$ .

$32 + 2 k - (\frac{1}{8} \cdot 256 + k \cdot - 2)$

Step 5.2.2.6

Combine $\frac{1}{8}$ and $256$ .

$32 + 2 k - (\frac{256}{8} + k \cdot - 2)$

Step 5.2.2.7

Cancel the common factor of $256$ and $8$ .

Tap for more steps...

Step 5.2.2.7.1

Factor $8$ out of $256$ .

$32 + 2 k - (\frac{8 \cdot 32}{8} + k \cdot - 2)$

Step 5.2.2.7.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.7.2.1

Factor $8$ out of $8$ .

$32 + 2 k - (\frac{8 \cdot 32}{8 (1)} + k \cdot - 2)$

Step 5.2.2.7.2.2

Cancel the common factor.

$32 + 2 k - (\frac{8 \cdot 32}{8 \cdot 1} + k \cdot - 2)$

Step 5.2.2.7.2.3

Rewrite the expression.

$32 + 2 k - (\frac{32}{1} + k \cdot - 2)$

Step 5.2.2.7.2.4

Divide $32$ by $1$ .

$32 + 2 k - (32 + k \cdot - 2)$

Step 5.2.2.8

Move $- 2$ to the left of $k$ .

$32 + 2 k - (32 - 2 k)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$32 + 2 k - 1 \cdot 32 - (- 2 k)$

Step 6.1.2

Multiply $- 1$ by $32$ .

$32 + 2 k - 32 - (- 2 k)$

Step 6.1.3

Multiply $- 2$ by $- 1$ .

$32 + 2 k - 32 + 2 k$

Step 6.2

Combine the opposite terms in $32 + 2 k - 32 + 2 k$ .

Tap for more steps...

Step 6.2.1

Subtract $32$ from $32$ .

$2 k + 0 + 2 k$

Step 6.2.2

Add $2 k$ and $0$ .

$2 k + 2 k$

Step 6.3

Add $2 k$ and $2 k$ .

$4 k$

Step 7