Calculus Examples

$\int_{0}^{\frac{π}{2}} \frac{\sin (x)}{1 + \cos^{2} (x)} d x$

Step 1

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- d u = \sin (x) d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = \cos (\frac{π}{2})$

Step 1.5

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

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{1}^{0} \frac{- 1}{1 + u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int_{1}^{0} - \frac{1}{1 + u^{2}} d u$

Step 3

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

$- \int_{1}^{0} \frac{1}{1 + u^{2}} d u$

Step 4

Rewrite $1$ as $1^{2}$ .

$- \int_{1}^{0} \frac{1}{1^{2} + u^{2}} d u$

Step 5

The integral of $\frac{1}{1^{2} + u^{2}}$ with respect to $u$ is $\arctan (u)]_{1}^{0}$ .

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

Step 6

Evaluate $\arctan (u)$ at $0$ and at $1$ .

$- (\arctan (0) - \arctan (1))$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

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

$- (0 - \arctan (1))$

Step 7.1.2

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$- (0 - \frac{π}{4})$

Step 7.2

Subtract $\frac{π}{4}$ from $0$ .

$- - \frac{π}{4}$

Step 7.3

Multiply $- - \frac{π}{4}$ .

Tap for more steps...

Step 7.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{π}{4}$

Step 7.3.2

Multiply $\frac{π}{4}$ by $1$ .

$\frac{π}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{4}$

Decimal Form:

$0.78539816 \dots$