Calculus Examples

$\int_{0}^{\frac{π}{2}} 3 \cos (x) \sin (\sin (x)) d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{\frac{π}{2}} \cos (x) \sin (\sin (x)) d x$

Step 2

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 2.1.2

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

$\cos (x)$

Step 2.2

Substitute the lower limit in for $x$ in $u = \sin (x)$ .

$u_{lower} = \sin (0)$

Step 2.3

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

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

Step 2.5

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

$u_{upper} = 1$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$3 \int_{0}^{1} \sin (u) d u$

Step 3

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

$3 (- \cos (u)]_{0}^{1})$

Step 4

Evaluate $- \cos (u)$ at $1$ and at $0$ .

$3 (- \cos (1) + \cos (0))$

Step 5

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

$3 (- \cos (1) + 1)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Evaluate $\cos (1)$ .

$3 (- 1 \cdot 0.5403023 + 1)$

Step 6.2

Multiply $- 1$ by $0.5403023$ .

$3 (- 0.5403023 + 1)$

Step 6.3

Add $- 0.5403023$ and $1$ .

$3 \cdot 0.45969769$

Step 6.4

Multiply $3$ by $0.45969769$ .

$1.37909308$