Calculus Examples

$\int_{1}^{2} x \sin (x^{2}) d x$

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 $u = x^{2}$ .

$u_{lower} = 1^{2}$

Step 1.3

One to any power is one.

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = 2^{2}$

Step 1.5

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

$u_{upper} = 4$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 4$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate $- \cos (u)$ at $4$ and at $1$ .

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Evaluate $\cos (4)$ .

$\frac{1}{2} (- - 0.65364362 + \cos (1))$

Step 6.1.2

Multiply $- 1$ by $- 0.65364362$ .

$\frac{1}{2} (0.65364362 + \cos (1))$

Step 6.1.3

Evaluate $\cos (1)$ .

$\frac{1}{2} (0.65364362 + 0.5403023)$

Step 6.2

Add $0.65364362$ and $0.5403023$ .

$\frac{1}{2} \cdot 1.19394592$

Step 6.3

Combine $\frac{1}{2}$ and $1.19394592$ .

$\frac{1.19394592}{2}$

Step 6.4

Divide $1.19394592$ by $2$ .

$0.59697296$