Calculus Examples

$\int_{0}^{π} x \cos (x) d x$

Step 1

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

$x \sin (x)]_{0}^{π} - \int_{0}^{π} \sin (x) d x$

Step 2

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

$x \sin (x)]_{0}^{π} - (- \cos (x)]_{0}^{π})$

Step 3

Simplify the answer.

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Step 3.1

Substitute and simplify.

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Step 3.1.1

Evaluate $x \sin (x)$ at $π$ and at $0$ .

$(π \sin (π)) + 0 \sin (0) - (- \cos (x)]_{0}^{π})$

Step 3.1.2

Evaluate $- \cos (x)$ at $π$ and at $0$ .

$(π \sin (π)) + 0 \sin (0) - ((- \cos (π)) + \cos (0))$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $0$ by $\sin (0)$ .

$π \sin (π) + 0 - ((- \cos (π)) + \cos (0))$

Step 3.1.3.2

Add $π \sin (π)$ and $0$ .

$π \sin (π) - (- \cos (π) + \cos (0))$

Step 3.2

The exact value of $\cos (0)$ is $1$ .

$π \sin (π) - (- \cos (π) + 1)$

Step 3.3

Simplify.

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Step 3.3.1

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

$π \sin (0) - (- \cos (π) + 1)$

Step 3.3.2

The exact value of $\sin (0)$ is $0$ .

$π \cdot 0 - (- \cos (π) + 1)$

Step 3.3.3

Multiply $π$ by $0$ .

$0 - (- \cos (π) + 1)$

Step 3.3.4

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

$0 - (- - \cos (0) + 1)$

Step 3.3.5

The exact value of $\cos (0)$ is $1$ .

$0 - (- (- 1 \cdot 1) + 1)$

Step 3.3.6

Multiply $- 1$ by $1$ .

$0 - (- - 1 + 1)$

Step 3.3.7

Multiply $- 1$ by $- 1$ .

$0 - (1 + 1)$

Step 3.3.8

Add $1$ and $1$ .

$0 - 1 \cdot 2$

Step 3.3.9

Multiply $- 1$ by $2$ .

$0 - 2$

Step 3.3.10

Subtract $2$ from $0$ .

$- 2$