Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{3}} (- 2 \sec^{2} (x) - 5 \cos (x)) d x$

Step 1

Remove parentheses.

$\int_{\frac{π}{4}}^{\frac{π}{3}} - 2 \sec^{2} (x) - 5 \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{\frac{π}{4}}^{\frac{π}{3}} - 2 \sec^{2} (x) d x + \int_{\frac{π}{4}}^{\frac{π}{3}} - 5 \cos (x) d x$

Step 3

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

$- 2 \int_{\frac{π}{4}}^{\frac{π}{3}} \sec^{2} (x) d x + \int_{\frac{π}{4}}^{\frac{π}{3}} - 5 \cos (x) d x$

Step 4

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$- 2 (\tan (x)]_{\frac{π}{4}}^{\frac{π}{3}}) + \int_{\frac{π}{4}}^{\frac{π}{3}} - 5 \cos (x) d x$

Step 5

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 2 (\tan (x)]_{\frac{π}{4}}^{\frac{π}{3}}) - 5 \int_{\frac{π}{4}}^{\frac{π}{3}} \cos (x) d x$

Step 6

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

$- 2 (\tan (x)]_{\frac{π}{4}}^{\frac{π}{3}}) - 5 (\sin (x)]_{\frac{π}{4}}^{\frac{π}{3}})$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Substitute and simplify.

Tap for more steps...

Step 7.1.1

Evaluate $\tan (x)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$- 2 ((\tan (\frac{π}{3})) - \tan (\frac{π}{4})) - 5 (\sin (x)]_{\frac{π}{4}}^{\frac{π}{3}})$

Step 7.1.2

Evaluate $\sin (x)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$- 2 (\tan (\frac{π}{3}) - \tan (\frac{π}{4})) - 5 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$- 2 (\sqrt{3} - \tan (\frac{π}{4})) - 5 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- 2 (\sqrt{3} - 1 \cdot 1) - 5 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2.3

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$- 2 (\sqrt{3} - 1 \cdot 1) - 5 (\frac{\sqrt{3}}{2} - \sin (\frac{π}{4}))$

Step 7.2.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- 2 (\sqrt{3} - 1 \cdot 1) - 5 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.2.5

Multiply $- 1$ by $1$ .

$- 2 (\sqrt{3} - 1) - 5 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3

Simplify.

Tap for more steps...

Step 7.3.1

Apply the distributive property.

$- 2 \sqrt{3} - 2 \cdot - 1 - 5 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.2

Multiply $- 2$ by $- 1$ .

$- 2 \sqrt{3} + 2 - 5 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.3

Apply the distributive property.

$- 2 \sqrt{3} + 2 - 5 \frac{\sqrt{3}}{2} - 5 (- \frac{\sqrt{2}}{2})$

Step 7.3.4

Combine $- 5$ and $\frac{\sqrt{3}}{2}$ .

$- 2 \sqrt{3} + 2 + \frac{- 5 \sqrt{3}}{2} - 5 (- \frac{\sqrt{2}}{2})$

Step 7.3.5

Multiply $- 5 (- \frac{\sqrt{2}}{2})$ .

Tap for more steps...

Step 7.3.5.1

Multiply $- 1$ by $- 5$ .

$- 2 \sqrt{3} + 2 + \frac{- 5 \sqrt{3}}{2} + 5 \frac{\sqrt{2}}{2}$

Step 7.3.5.2

Combine $5$ and $\frac{\sqrt{2}}{2}$ .

$- 2 \sqrt{3} + 2 + \frac{- 5 \sqrt{3}}{2} + \frac{5 \sqrt{2}}{2}$

Step 7.3.6

Move the negative in front of the fraction.

$- 2 \sqrt{3} + 2 - \frac{5 \sqrt{3}}{2} + \frac{5 \sqrt{2}}{2}$

Step 7.3.7

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

$- 2 \sqrt{3} \cdot \frac{2}{2} - \frac{5 \sqrt{3}}{2} + 2 + \frac{5 \sqrt{2}}{2}$

Step 7.3.8

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

$\frac{- 2 \sqrt{3} \cdot 2}{2} - \frac{5 \sqrt{3}}{2} + 2 + \frac{5 \sqrt{2}}{2}$

Step 7.3.9

Combine the numerators over the common denominator.

$\frac{- 2 \sqrt{3} \cdot 2 - 5 \sqrt{3}}{2} + 2 + \frac{5 \sqrt{2}}{2}$

Step 7.3.10

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

$\frac{- 2 \sqrt{3} \cdot 2 - 5 \sqrt{3}}{2} + 2 \cdot \frac{2}{2} + \frac{5 \sqrt{2}}{2}$

Step 7.3.11

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

$\frac{- 2 \sqrt{3} \cdot 2 - 5 \sqrt{3}}{2} + \frac{2 \cdot 2}{2} + \frac{5 \sqrt{2}}{2}$

Step 7.3.12

Combine the numerators over the common denominator.

$\frac{- 2 \sqrt{3} \cdot 2 - 5 \sqrt{3} + 2 \cdot 2}{2} + \frac{5 \sqrt{2}}{2}$

Step 7.3.13

Combine the numerators over the common denominator.

$\frac{- 2 \sqrt{3} \cdot 2 - 5 \sqrt{3} + 2 \cdot 2 + 5 \sqrt{2}}{2}$

Step 7.3.14

Multiply $2$ by $- 2$ .

$\frac{- 4 \sqrt{3} - 5 \sqrt{3} + 2 \cdot 2 + 5 \sqrt{2}}{2}$

Step 7.3.15

Multiply $2$ by $2$ .

$\frac{- 4 \sqrt{3} - 5 \sqrt{3} + 4 + 5 \sqrt{2}}{2}$

Step 7.3.16

Subtract $5 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$\frac{- 9 \sqrt{3} + 4 + 5 \sqrt{2}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{- 9 \sqrt{3} + 4 + 5 \sqrt{2}}{2}$

Decimal Form:

$- 2.25869472 \dots$