Calculus Examples

$\int_{3}^{5} x \cos (x) d x$

Step 1

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

$x \sin (x)]_{3}^{5} - \int_{3}^{5} \sin (x) d x$

Step 2

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

$x \sin (x)]_{3}^{5} - (- \cos (x)]_{3}^{5})$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $x \sin (x)$ at $5$ and at $3$ .

$(5 \sin (5)) - 3 \sin (3) - (- \cos (x)]_{3}^{5})$

Step 3.2

Evaluate $- \cos (x)$ at $5$ and at $3$ .

$(5 \sin (5)) - 3 \sin (3) - ((- \cos (5)) + \cos (3))$

Step 3.3

Remove parentheses.

$5 \sin (5) - 3 \sin (3) - (- \cos (5) + \cos (3))$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Evaluate $\sin (5)$ .

$5 \cdot - 0.95892427 - 3 \sin (3) - (- \cos (5) + \cos (3))$

Step 4.1.2

Multiply $5$ by $- 0.95892427$ .

$- 4.79462137 - 3 \sin (3) - (- \cos (5) + \cos (3))$

Step 4.1.3

Evaluate $\sin (3)$ .

$- 4.79462137 - 3 \cdot 0.14112 - (- \cos (5) + \cos (3))$

Step 4.1.4

Multiply $- 3$ by $0.14112$ .

$- 4.79462137 - 0.42336002 - (- \cos (5) + \cos (3))$

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

Evaluate $\cos (5)$ .

$- 4.79462137 - 0.42336002 - (- 1 \cdot 0.28366218 + \cos (3))$

Step 4.1.5.2

Multiply $- 1$ by $0.28366218$ .

$- 4.79462137 - 0.42336002 - (- 0.28366218 + \cos (3))$

Step 4.1.5.3

Evaluate $\cos (3)$ .

$- 4.79462137 - 0.42336002 - (- 0.28366218 - 0.98999249)$

Step 4.1.6

Subtract $0.98999249$ from $- 0.28366218$ .

$- 4.79462137 - 0.42336002 - - 1.27365468$

Step 4.1.7

Multiply $- 1$ by $- 1.27365468$ .

$- 4.79462137 - 0.42336002 + 1.27365468$

Step 4.2

Subtract $0.42336002$ from $- 4.79462137$ .

$- 5.21798139 + 1.27365468$

Step 4.3

Add $- 5.21798139$ and $1.27365468$ .

$- 3.94432671$