Calculus Examples

$\int_{0}^{\frac{π}{8}} \cos^{4} (2 x) d x$

Step 1

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

Tap for more steps...

Step 1.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u_{1} = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 1.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = 2 x$ .

$u_{upper} = 2 \frac{π}{8}$

Step 1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.1

Factor $2$ out of $8$ .

$u_{upper} = 2 \frac{π}{2 (4)}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 4}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{4}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{4}$

Step 1.7

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

$\int_{0}^{\frac{π}{4}} \cos^{4} (u_{1}) \frac{1}{2} d u_{1}$

Step 2

Combine $\cos^{4} (u_{1})$ and $\frac{1}{2}$ .

$\int_{0}^{\frac{π}{4}} \frac{\cos^{4} (u_{1})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int_{0}^{\frac{π}{4}} \cos^{4} (u_{1}) d u_{1}$

Step 4

Simplify with factoring out.

Tap for more steps...

Step 4.1

Factor $2$ out of $4$ .

$\frac{1}{2} \int_{0}^{\frac{π}{4}} {\cos (u_{1})}^{2 (2)} d u_{1}$

Step 4.2

Rewrite ${\cos (u_{1})}^{2 (2)}$ as exponentiation.

$\frac{1}{2} \int_{0}^{\frac{π}{4}} {(\cos^{2} (u_{1}))}^{2} d u_{1}$

Step 5

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$\frac{1}{2} \int_{0}^{\frac{π}{4}} {(\frac{1 + \cos (2 u_{1})}{2})}^{2} d u_{1}$

Step 6

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

Tap for more steps...

Step 6.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 6.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 6.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 6.1.3

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

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 6.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 6.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 \frac{π}{4}$

Step 6.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 6.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 6.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 6.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 6.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

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

Step 7

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

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

Step 8

Simplify terms.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 8.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{0}^{\frac{π}{2}} {(\frac{1 + \cos (u_{2})}{2})}^{2} d u_{2}$

Step 8.2

Rewrite $\frac{1 + \cos (u_{2})}{2}$ as a product.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} {(\frac{1}{2} \cdot (1 + \cos (u_{2})))}^{2} d u_{2}$

Step 8.3

Expand ${(\frac{1}{2} \cdot (1 + \cos (u_{2})))}^{2}$ .

Tap for more steps...

Step 8.3.1

Rewrite the exponentiation as a product.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} \frac{1}{2} \cdot (1 + \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2}))) d u_{2}$

Step 8.3.2

Apply the distributive property.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2}))) d u_{2}$

Step 8.3.3

Apply the distributive property.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) d u_{2}$

Step 8.3.4

Apply the distributive property.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) d u_{2}$

Step 8.3.5

Apply the distributive property.

$\frac{1}{4} \int_{0}^{\frac{π}{2}} \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) d u_{2}$

Step 8.3.6

Apply the distributive property.

Step 8.3.7