Calculus Examples

$\int_{0}^{0.9} \cos (\sqrt{6 x}) d x$

Step 1

Let $u_{1} = 6 x$ . Then $d u_{1} = 6 d x$ , so $\frac{1}{6} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 1.1

Let $u_{1} = 6 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 1.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 1.1.3

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

$6 \cdot 1$

Step 1.1.4

Multiply $6$ by $1$ .

$6$

Step 1.2

Substitute the lower limit in for $x$ in $u_{1} = 6 x$ .

$u_{lower} = 6 \cdot 0$

Step 1.3

Multiply $6$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = 6 x$ .

$u_{upper} = 6 \cdot 0.9$

Step 1.5

Multiply $6$ by $0.9$ .

$u_{upper} = 5.4$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 5.4$

Step 1.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{0}^{5.4} \cos (\sqrt{u_{1}}) \frac{1}{6} d u_{1}$

Step 2

Combine $\cos (\sqrt{u_{1}})$ and $\frac{1}{6}$ .

$\int_{0}^{5.4} \frac{\cos (\sqrt{u_{1}})}{6} d u_{1}$

Step 3

Since $\frac{1}{6}$ is constant with respect to $u_{1}$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int_{0}^{5.4} \cos (\sqrt{u_{1}}) d u_{1}$

Step 4

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

$\frac{1}{6} \int_{0}^{\sqrt{5.4}} 2 u_{2} \cos (u_{2}) d u_{2}$

Step 5

Since $2$ is constant with respect to $u_{2}$ , move $2$ out of the integral.

$\frac{1}{6} (2 \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Combine $2$ and $\frac{1}{6}$ .

$\frac{2}{6} \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2}$

Step 6.2

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 6.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2}$

Step 6.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2}$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2}$

Step 6.2.2.3

Rewrite the expression.

$\frac{1}{3} \int_{0}^{\sqrt{5.4}} u_{2} \cos (u_{2}) d u_{2}$

Step 7

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u_{2}$ and $dv = \cos (u_{2})$ .

$\frac{1}{3} (u_{2} \sin (u_{2})]_{0}^{\sqrt{5.4}} - \int_{0}^{\sqrt{5.4}} \sin (u_{2}) d u_{2})$

Step 8

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

$\frac{1}{3} (u_{2} \sin (u_{2})]_{0}^{\sqrt{5.4}} - (- \cos (u_{2})]_{0}^{\sqrt{5.4}}))$

Step 9

Substitute and simplify.

Tap for more steps...

Step 9.1

Evaluate $u_{2} \sin (u_{2})$ at $\sqrt{5.4}$ and at $0$ .

$\frac{1}{3} ((\sqrt{5.4} \sin (\sqrt{5.4})) + 0 \sin (0) - (- \cos (u_{2})]_{0}^{\sqrt{5.4}}))$

Step 9.2

Evaluate $- \cos (u_{2})$ at $\sqrt{5.4}$ and at $0$ .

$\frac{1}{3} ((\sqrt{5.4} \sin (\sqrt{5.4})) + 0 \sin (0) - ((- \cos (\sqrt{5.4})) + \cos (0)))$

Step 9.3

Simplify.

Tap for more steps...

Step 9.3.1

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

$\frac{1}{3} (\sqrt{5.4} \sin (\sqrt{5.4}) + 0 - ((- \cos (\sqrt{5.4})) + \cos (0)))$

Step 9.3.2

Add $\sqrt{5.4} \sin (\sqrt{5.4})$ and $0$ .

$\frac{1}{3} (\sqrt{5.4} \sin (\sqrt{5.4}) - (- \cos (\sqrt{5.4}) + \cos (0)))$

Step 10

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

$\frac{1}{3} (\sqrt{5.4} \sin (\sqrt{5.4}) - (- \cos (\sqrt{5.4}) + 1))$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Evaluate $\sin (\sqrt{5.4})$ .

$\frac{1}{3} (\sqrt{5.4} \cdot 0.72964498 - (- \cos (\sqrt{5.4}) + 1))$

Step 11.2

Multiply $\sqrt{5.4}$ by $0.72964498$ .

$\frac{1}{3} (1.69554172 - (- \cos (\sqrt{5.4}) + 1))$

Step 11.3

Evaluate $\cos (\sqrt{5.4})$ .

$\frac{1}{3} (1.69554172 - (- - 0.68382614 + 1))$

Step 11.4

Multiply $- 1$ by $- 0.68382614$ .

$\frac{1}{3} (1.69554172 - (0.68382614 + 1))$

Step 11.5

Add $0.68382614$ and $1$ .

$\frac{1}{3} (1.69554172 - 1 \cdot 1.68382614)$

Step 11.6

Multiply $- 1$ by $1.68382614$ .

$\frac{1}{3} (1.69554172 - 1.68382614)$

Step 11.7

Subtract $1.68382614$ from $1.69554172$ .

$\frac{1}{3} \cdot 0.01171557$

Step 11.8

Combine $\frac{1}{3}$ and $0.01171557$ .

$\frac{0.01171557}{3}$

Step 11.9

Divide $0.01171557$ by $3$ .

$0.00390519$