Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{2}} x + \cos (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{\frac{π}{6}}^{\frac{π}{2}} x d x + \int_{\frac{π}{6}}^{\frac{π}{2}} \cos (x) d x$

Step 2

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

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

Step 3

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$\frac{1}{2} x^{2}]_{\frac{π}{6}}^{\frac{π}{2}} + \sin (x)]_{\frac{π}{6}}^{\frac{π}{2}}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Combine $\frac{1}{2} x^{2}]_{\frac{π}{6}}^{\frac{π}{2}}$ and $\sin (x)]_{\frac{π}{6}}^{\frac{π}{2}}$ .

$\frac{1}{2} x^{2} + \sin (x)]_{\frac{π}{6}}^{\frac{π}{2}}$

Step 4.2

Evaluate $\frac{1}{2} x^{2} + \sin (x)$ at $\frac{π}{2}$ and at $\frac{π}{6}$ .

$\frac{1}{2} {(\frac{π}{2})}^{2} + \sin (\frac{π}{2}) - (\frac{1}{2} {(\frac{π}{6})}^{2} + \sin (\frac{π}{6}))$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{2} {(\frac{π}{2})}^{2} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \sin (\frac{π}{6}))$

Step 4.3.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1}{2} {(\frac{π}{2})}^{2} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Apply the product rule to $\frac{π}{2}$ .

$\frac{1}{2} \cdot \frac{π^{2}}{2^{2}} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4.2

Combine.

$\frac{1 π^{2}}{2 \cdot 2^{2}} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4.3

Multiply $π^{2}$ by $1$ .

$\frac{π^{2}}{2 \cdot 2^{2}} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4.4

Multiply $2$ by $2^{2}$ by adding the exponents.

Tap for more steps...

Step 4.4.4.1

Multiply $2$ by $2^{2}$ .

Tap for more steps...

Step 4.4.4.1.1

Raise $2$ to the power of $1$ .

$\frac{π^{2}}{2^{1} \cdot 2^{2}} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4.4.1.2

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

$\frac{π^{2}}{2^{1 + 2}} + 1 - (\frac{1}{2} {(\frac{π}{6})}^{2} + \frac{1}{2})$

Step 4.4.4.2

Add $1$ and $2$ .

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

Step 4.4.5

Apply the product rule to $\frac{π}{6}$ .

$\frac{π^{2}}{2^{3}} + 1 - (\frac{1}{2} \cdot \frac{π^{2}}{6^{2}} + \frac{1}{2})$

Step 4.4.6

Combine.

$\frac{π^{2}}{2^{3}} + 1 - (\frac{1 π^{2}}{2 \cdot 6^{2}} + \frac{1}{2})$

Step 4.4.7

Multiply $π^{2}$ by $1$ .

$\frac{π^{2}}{2^{3}} + 1 - (\frac{π^{2}}{2 \cdot 6^{2}} + \frac{1}{2})$

Step 4.4.8

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

$\frac{π^{2}}{2^{3}} + 1 - (\frac{π^{2}}{2 \cdot 36} + \frac{1}{2})$

Step 4.4.9

Multiply $2$ by $36$ .

$\frac{π^{2}}{2^{3}} + 1 - (\frac{π^{2}}{72} + \frac{1}{2})$

Step 4.4.10

Apply the distributive property.

$\frac{π^{2}}{2^{3}} + 1 - \frac{π^{2}}{72} - \frac{1}{2}$

Step 4.4.11

Raise $2$ to the power of $3$ .

$\frac{π^{2}}{8} + 1 - \frac{π^{2}}{72} - \frac{1}{2}$

Step 4.4.12

To write $\frac{π^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{π^{2}}{8} \cdot \frac{9}{9} - \frac{π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.13

Write each expression with a common denominator of $72$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.4.13.1

Multiply $\frac{π^{2}}{8}$ by $\frac{9}{9}$ .

$\frac{π^{2} \cdot 9}{8 \cdot 9} - \frac{π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.13.2

Multiply $8$ by $9$ .

$\frac{π^{2} \cdot 9}{72} - \frac{π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.14

Combine the numerators over the common denominator.

$\frac{π^{2} \cdot 9 - π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.15

Subtract $π^{2}$ from $π^{2} \cdot 9$ .

Tap for more steps...

Step 4.4.15.1

Reorder $π^{2}$ and $9$ .

$\frac{9 \cdot π^{2} - π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.15.2

Subtract $π^{2}$ from $9 \cdot π^{2}$ .

$\frac{8 \cdot π^{2}}{72} + 1 - \frac{1}{2}$

Step 4.4.16

Write $1$ as a fraction with a common denominator.

$\frac{8 \cdot π^{2}}{72} + \frac{2}{2} - \frac{1}{2}$

Step 4.4.17

Combine the numerators over the common denominator.

$\frac{8 \cdot π^{2}}{72} + \frac{2 - 1}{2}$

Step 4.4.18

Subtract $1$ from $2$ .

$\frac{8 \cdot π^{2}}{72} + \frac{1}{2}$

Step 5

Cancel the common factor of $8$ and $72$ .

Tap for more steps...

Step 5.1

Factor $8$ out of $8 \cdot π^{2}$ .

$\frac{8 (π^{2})}{72} + \frac{1}{2}$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $8$ out of $72$ .

$\frac{8 π^{2}}{8 \cdot 9} + \frac{1}{2}$

Step 5.2.2

Cancel the common factor.

$\frac{8 π^{2}}{8 \cdot 9} + \frac{1}{2}$

Step 5.2.3

Rewrite the expression.

$\frac{π^{2}}{9} + \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{9} + \frac{1}{2}$

Decimal Form:

$1.59662271 \dots$