Calculus Examples

$\int_{\frac{π}{2}}^{\frac{3 π}{2}} x \sin (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (x)$ .

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

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Multiply $\int_{\frac{π}{2}}^{\frac{3 π}{2}} \cos (x) d x$ by $1$ .

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

Step 4

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

$x (- \cos (x))]_{\frac{π}{2}}^{\frac{3 π}{2}} + \sin (x)]_{\frac{π}{2}}^{\frac{3 π}{2}}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Combine $x (- \cos (x))]_{\frac{π}{2}}^{\frac{3 π}{2}}$ and $\sin (x)]_{\frac{π}{2}}^{\frac{3 π}{2}}$ .

$x (- \cos (x)) + \sin (x)]_{\frac{π}{2}}^{\frac{3 π}{2}}$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $x (- \cos (x)) + \sin (x)$ at $\frac{3 π}{2}$ and at $\frac{π}{2}$ .

$(\frac{3 π}{2} (- \cos (\frac{3 π}{2})) + \sin (\frac{3 π}{2})) - (\frac{π}{2} (- \cos (\frac{π}{2})) + \sin (\frac{π}{2}))$

Step 5.2.2

Simplify.

Tap for more steps...

Step 5.2.2.1

Combine $\frac{3 π}{2}$ and $\cos (\frac{3 π}{2})$ .

$- \frac{3 π \cos (\frac{3 π}{2})}{2} + \sin (\frac{3 π}{2}) - (\frac{π}{2} (- \cos (\frac{π}{2})) + \sin (\frac{π}{2}))$

Step 5.2.2.2

Combine $\frac{π}{2}$ and $\cos (\frac{π}{2})$ .

$- \frac{3 π \cos (\frac{3 π}{2})}{2} + \sin (\frac{3 π}{2}) - (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2}))$

Step 5.2.2.3

To write $- (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2}))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 π \cos (\frac{3 π}{2})}{2} - (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2})) \cdot \frac{2}{2} + \sin (\frac{3 π}{2})$

Step 5.2.2.4

Combine $- (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2}))$ and $\frac{2}{2}$ .

$- \frac{3 π \cos (\frac{3 π}{2})}{2} + \frac{- (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2})) \cdot 2}{2} + \sin (\frac{3 π}{2})$

Step 5.2.2.5

Combine the numerators over the common denominator.

$\frac{- 3 π \cos (\frac{3 π}{2}) - (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2})) \cdot 2}{2} + \sin (\frac{3 π}{2})$

Step 5.2.2.6

Multiply $2$ by $- 1$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{π \cos (\frac{π}{2})}{2} + \sin (\frac{π}{2}))}{2} + \sin (\frac{3 π}{2})$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{π \cdot 0}{2} + \sin (\frac{π}{2}))}{2} + \sin (\frac{3 π}{2})$

Step 5.3.2

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

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{π \cdot 0}{2} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.3

Multiply $π$ by $0$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{0}{2} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.4

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 5.3.4.1

Factor $2$ out of $0$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{2 (0)}{2} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.4.2

Cancel the common factors.

Tap for more steps...

Step 5.3.4.2.1

Factor $2$ out of $2$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{2 \cdot 0}{2 \cdot 1} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.4.2.2

Cancel the common factor.

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{2 \cdot 0}{2 \cdot 1} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.4.2.3

Rewrite the expression.

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- \frac{0}{1} + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.4.2.4

Divide $0$ by $1$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (- 0 + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.5

Multiply $- 1$ by $0$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 (0 + 1)}{2} + \sin (\frac{3 π}{2})$

Step 5.3.6

Add $0$ and $1$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 \cdot 1}{2} + \sin (\frac{3 π}{2})$

Step 5.3.7

Multiply $- 2$ by $1$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2}{2} + \sin (\frac{3 π}{2})$

Step 5.3.8

To write $\sin (\frac{3 π}{2})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2}{2} + \sin (\frac{3 π}{2}) \cdot \frac{2}{2}$

Step 5.3.9

Combine $\sin (\frac{3 π}{2})$ and $\frac{2}{2}$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2}{2} + \frac{\sin (\frac{3 π}{2}) \cdot 2}{2}$

Step 5.3.10

Combine the numerators over the common denominator.

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 + \sin (\frac{3 π}{2}) \cdot 2}{2}$

Step 5.3.11

Move $2$ to the left of $\sin (\frac{3 π}{2})$ .

$\frac{- 3 π \cos (\frac{3 π}{2}) - 2 + 2 \cdot \sin (\frac{3 π}{2})}{2}$

Step 5.3.12

Factor $- 1$ out of $- 3 π \cos (\frac{3 π}{2})$ .

$\frac{- (3 π \cos (\frac{3 π}{2})) - 2 + 2 \sin (\frac{3 π}{2})}{2}$

Step 5.3.13

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (3 π \cos (\frac{3 π}{2})) - 1 (2) + 2 \sin (\frac{3 π}{2})}{2}$

Step 5.3.14

Factor $- 1$ out of $- (3 π \cos (\frac{3 π}{2})) - 1 (2)$ .

$\frac{- (3 π \cos (\frac{3 π}{2}) + 2) + 2 \sin (\frac{3 π}{2})}{2}$

Step 5.3.15

Factor $- 1$ out of $2 \sin (\frac{3 π}{2})$ .

$\frac{- (3 π \cos (\frac{3 π}{2}) + 2) - (- 2 \sin (\frac{3 π}{2}))}{2}$

Step 5.3.16

Factor $- 1$ out of $- (3 π \cos (\frac{3 π}{2}) + 2) - (- 2 \sin (\frac{3 π}{2}))$ .

$\frac{- (3 π \cos (\frac{3 π}{2}) + 2 - 2 \sin (\frac{3 π}{2}))}{2}$

Step 5.3.17

Rewrite $- (3 π \cos (\frac{3 π}{2}) + 2 - 2 \sin (\frac{3 π}{2}))$ as $- 1 (3 π \cos (\frac{3 π}{2}) + 2 - 2 \sin (\frac{3 π}{2}))$ .

$\frac{- 1 (3 π \cos (\frac{3 π}{2}) + 2 - 2 \sin (\frac{3 π}{2}))}{2}$

Step 5.3.18

Move the negative in front of the fraction.

$- \frac{3 π \cos (\frac{3 π}{2}) + 2 - 2 \sin (\frac{3 π}{2})}{2}$

Step 5.4

Simplify.

Tap for more steps...

Step 5.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{3 π \cos (\frac{π}{2}) + 2 - 2 \sin (\frac{3 π}{2})}{2}$

Step 5.4.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- \frac{3 π \cdot 0 + 2 - 2 \sin (\frac{3 π}{2})}{2}$

Step 5.4.3

Multiply $3 π \cdot 0$ .

Tap for more steps...

Step 5.4.3.1

Multiply $0$ by $3$ .

$- \frac{0 π + 2 - 2 \sin (\frac{3 π}{2})}{2}$

Step 5.4.3.2

Multiply $0$ by $π$ .

$- \frac{0 + 2 - 2 \sin (\frac{3 π}{2})}{2}$

Step 5.4.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \frac{0 + 2 - 2 (- \sin (\frac{π}{2}))}{2}$

Step 5.4.5

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

$- \frac{0 + 2 - 2 (- 1 \cdot 1)}{2}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$- \frac{0 + 2 - 2 \cdot - 1}{2}$

Step 5.4.7

Multiply $- 2$ by $- 1$ .

$- \frac{0 + 2 + 2}{2}$

Step 5.4.8

Cancel the common factor of $0 + 2 + 2$ and $2$ .

Tap for more steps...

Step 5.4.8.1

Factor $2$ out of $0$ .

$- \frac{2 \cdot 0 + 2 + 2}{2}$

Step 5.4.8.2

Factor $2$ out of $2$ .

$- \frac{2 \cdot 0 + 2 \cdot 1 + 2}{2}$

Step 5.4.8.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot 1$ .

$- \frac{2 \cdot (0 + 1) + 2}{2}$

Step 5.4.8.4

Factor $2$ out of $2$ .

$- \frac{2 \cdot (0 + 1) + 2 \cdot 1}{2}$

Step 5.4.8.5

Factor $2$ out of $2 \cdot (0 + 1) + 2 (1)$ .

$- \frac{2 \cdot (0 + 1 + 1)}{2}$

Step 5.4.8.6

Cancel the common factors.

Tap for more steps...

Step 5.4.8.6.1

Factor $2$ out of $2$ .

$- \frac{2 \cdot (0 + 1 + 1)}{2 (1)}$

Step 5.4.8.6.2

Cancel the common factor.

$- \frac{2 \cdot (0 + 1 + 1)}{2 \cdot 1}$

Step 5.4.8.6.3

Rewrite the expression.

$- \frac{0 + 1 + 1}{1}$

Step 5.4.8.6.4

Divide $0 + 1 + 1$ by $1$ .

$- (0 + 1 + 1)$

Step 5.4.9

Add $0$ and $1$ .

$- (1 + 1)$

Step 5.4.10

Add $1$ and $1$ .

$- 1 \cdot 2$

Step 5.4.11

Multiply $- 1$ by $2$ .

$- 2$