Calculus Examples

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

Step 1

Let $u = x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2}$ .

$u_{lower} = 0^{2}$

Step 1.3

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2}$ .

$u_{upper} = {(\frac{π}{2})}^{2}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Apply the product rule to $\frac{π}{2}$ .

$u_{upper} = \frac{π^{2}}{2^{2}}$

Step 1.5.2

Raise $2$ to the power of $2$ .

$u_{upper} = \frac{π^{2}}{4}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π^{2}}{4}$

Step 1.7

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

$\int_{0}^{\frac{π^{2}}{4}} \sin (u) \frac{1}{2} d u$

Step 2

Combine $\sin (u)$ and $\frac{1}{2}$ .

$\int_{0}^{\frac{π^{2}}{4}} \frac{\sin (u)}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{0}^{\frac{π^{2}}{4}} \sin (u) d u$

Step 4

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

$\frac{1}{2} - \cos (u)]_{0}^{\frac{π^{2}}{4}}$

Step 5

Evaluate $- \cos (u)$ at $\frac{π^{2}}{4}$ and at $0$ .

$\frac{1}{2} (- \cos (\frac{π^{2}}{4}) + \cos (0))$

Step 6

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

$\frac{1}{2} (- \cos (\frac{π^{2}}{4}) + 1)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Evaluate $\cos (\frac{π^{2}}{4})$ .

$\frac{1}{2} (- - 0.78121189 + 1)$

Step 7.2

Multiply $- 1$ by $- 0.78121189$ .

$\frac{1}{2} (0.78121189 + 1)$

Step 7.3

Add $0.78121189$ and $1$ .

$\frac{1}{2} \cdot 1.78121189$

Step 7.4

Combine $\frac{1}{2}$ and $1.78121189$ .

$\frac{1.78121189}{2}$

Step 7.5

Divide $1.78121189$ by $2$ .

$0.89060594$