Calculus Examples

$\int_{0}^{4} y \cos (y) d y$

Step 1

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

$y \sin (y)]_{0}^{4} - \int_{0}^{4} \sin (y) d y$

Step 2

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

$y \sin (y)]_{0}^{4} - (- \cos (y)]_{0}^{4})$

Step 3

Simplify the answer.

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Step 3.1

Substitute and simplify.

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Step 3.1.1

Evaluate $y \sin (y)$ at $4$ and at $0$ .

$(4 \sin (4)) + 0 \sin (0) - (- \cos (y)]_{0}^{4})$

Step 3.1.2

Evaluate $- \cos (y)$ at $4$ and at $0$ .

$(4 \sin (4)) + 0 \sin (0) - ((- \cos (4)) + \cos (0))$

Step 3.1.3

Simplify.

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Step 3.1.3.1

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

$4 \sin (4) + 0 - ((- \cos (4)) + \cos (0))$

Step 3.1.3.2

Add $4 \sin (4)$ and $0$ .

$4 \sin (4) - (- \cos (4) + \cos (0))$

Step 3.2

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

$4 \sin (4) - (- \cos (4) + 1)$

Step 3.3

Simplify.

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Step 3.3.1

Evaluate $\sin (4)$ .

$4 \cdot - 0.75680249 - (- \cos (4) + 1)$

Step 3.3.2

Multiply $4$ by $- 0.75680249$ .

$- 3.02720998 - (- \cos (4) + 1)$

Step 3.3.3

Evaluate $\cos (4)$ .

$- 3.02720998 - (- - 0.65364362 + 1)$

Step 3.3.4

Multiply $- 1$ by $- 0.65364362$ .

$- 3.02720998 - (0.65364362 + 1)$

Step 3.3.5

Add $0.65364362$ and $1$ .

$- 3.02720998 - 1 \cdot 1.65364362$

Step 3.3.6

Multiply $- 1$ by $1.65364362$ .

$- 3.02720998 - 1.65364362$

Step 3.3.7

Subtract $1.65364362$ from $- 3.02720998$ .

$- 4.6808536$