Calculus Examples

$\int_{0}^{\frac{π}{2}} \frac{2 \sin (2 t)}{5 - \cos (2 t)} d t$

Step 1

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$2 \int_{0}^{\frac{π}{2}} \frac{\sin (2 t)}{5 - \cos (2 t)} d t$

Step 2

Let $u_{2} = 5 - \cos (2 t)$ . Then $d u_{2} = 2 \sin (2 t) d t$ , so $\frac{1}{2} d u_{2} = \sin (2 t) d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.1

Let $u_{2} = 5 - \cos (2 t)$ . Find $\frac{d u_{2}}{d t}$ .

Tap for more steps...

Step 2.1.1

Differentiate $5 - \cos (2 t)$ .

$\frac{d}{d t} [5 - \cos (2 t)]$

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $5 - \cos (2 t)$ with respect to $t$ is $\frac{d}{d t} [5] + \frac{d}{d t} [- \cos (2 t)]$ .

$\frac{d}{d t} [5] + \frac{d}{d t} [- \cos (2 t)]$

Step 2.1.2.2

Since $5$ is constant with respect to $t$ , the derivative of $5$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \cos (2 t)]$

Step 2.1.3

Evaluate $\frac{d}{d t} [- \cos (2 t)]$ .

Tap for more steps...

Step 2.1.3.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- \cos (2 t)$ with respect to $t$ is $- \frac{d}{d t} [\cos (2 t)]$ .

$0 - \frac{d}{d t} [\cos (2 t)]$

Step 2.1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 2 t$ .

Tap for more steps...

Step 2.1.3.2.1

To apply the Chain Rule, set $u_{1}$ as $2 t$ .

$0 - (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [2 t])$

Step 2.1.3.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$0 - (- \sin (u_{1}) \frac{d}{d t} [2 t])$

Step 2.1.3.2.3

Replace all occurrences of $u_{1}$ with $2 t$ .

$0 - (- \sin (2 t) \frac{d}{d t} [2 t])$

Step 2.1.3.3

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

$0 - (- \sin (2 t) (2 \frac{d}{d t} [t]))$

Step 2.1.3.4

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

$0 - (- \sin (2 t) (2 \cdot 1))$

Step 2.1.3.5

Multiply $2$ by $1$ .

$0 - (- \sin (2 t) \cdot 2)$

Step 2.1.3.6

Multiply $2$ by $- 1$ .

$0 - (- 2 \sin (2 t))$

Step 2.1.3.7

Multiply $- 2$ by $- 1$ .

$0 + 2 \sin (2 t)$

Step 2.1.4

Add $0$ and $2 \sin (2 t)$ .

$2 \sin (2 t)$

Step 2.2

Substitute the lower limit in for $t$ in $u_{2} = 5 - \cos (2 t)$ .

$u_{lower} = 5 - \cos (2 \cdot 0)$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $2$ by $0$ .

$u_{lower} = 5 - \cos (0)$

Step 2.3.1.2

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

$u_{lower} = 5 - 1 \cdot 1$

Step 2.3.1.3

Multiply $- 1$ by $1$ .

$u_{lower} = 5 - 1$

Step 2.3.2

Subtract $1$ from $5$ .

$u_{lower} = 4$

Step 2.4

Substitute the upper limit in for $t$ in $u_{2} = 5 - \cos (2 t)$ .

$u_{upper} = 5 - \cos (2 \frac{π}{2})$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.1.1.1

Cancel the common factor.

$u_{upper} = 5 - \cos (2 \frac{π}{2})$

Step 2.5.1.1.2

Rewrite the expression.

$u_{upper} = 5 - \cos (π)$

Step 2.5.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$u_{upper} = 5 - - \cos (0)$

Step 2.5.1.3

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

$u_{upper} = 5 - (- 1 \cdot 1)$

Step 2.5.1.4

Multiply $- (- 1 \cdot 1)$ .

Tap for more steps...

Step 2.5.1.4.1

Multiply $- 1$ by $1$ .

$u_{upper} = 5 - - 1$

Step 2.5.1.4.2

Multiply $- 1$ by $- 1$ .

$u_{upper} = 5 + 1$

Step 2.5.2

Add $5$ and $1$ .

$u_{upper} = 6$

Step 2.6

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

$u_{lower} = 4$

$u_{upper} = 6$

Step 2.7

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

$2 \int_{4}^{6} \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

$2 \int_{4}^{6} \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 3.2

Move $2$ to the left of $u_{2}$ .

$2 \int_{4}^{6} \frac{1}{2 u_{2}} d u_{2}$

Step 4

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

$2 (\frac{1}{2} \int_{4}^{6} \frac{1}{u_{2}} d u_{2})$

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

$\frac{2}{2} \int_{4}^{6} \frac{1}{u_{2}} d u_{2}$

Step 5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{2}{2} \int_{4}^{6} \frac{1}{u_{2}} d u_{2}$

Step 5.2.2

Rewrite the expression.

$1 \int_{4}^{6} \frac{1}{u_{2}} d u_{2}$

Step 5.3

Multiply $\int_{4}^{6} \frac{1}{u_{2}} d u_{2}$ by $1$ .

$\int_{4}^{6} \frac{1}{u_{2}} d u_{2}$

Step 6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| u_{2} |)]_{4}^{6}$

Step 7

Evaluate $\ln (| u_{2} |)$ at $6$ and at $4$ .

$\ln (| 6 |) - \ln (| 4 |)$

Step 8

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 6 |}{| 4 |})$

Step 9

Simplify.

Tap for more steps...

Step 9.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\ln (\frac{6}{| 4 |})$

Step 9.2

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\ln (\frac{6}{4})$

Step 9.3

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 9.3.1

Factor $2$ out of $6$ .

$\ln (\frac{2 (3)}{4})$

Step 9.3.2

Cancel the common factors.

Tap for more steps...

Step 9.3.2.1

Factor $2$ out of $4$ .

$\ln (\frac{2 \cdot 3}{2 \cdot 2})$

Step 9.3.2.2

Cancel the common factor.

$\ln (\frac{2 \cdot 3}{2 \cdot 2})$

Step 9.3.2.3

Rewrite the expression.

$\ln (\frac{3}{2})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{3}{2})$

Decimal Form:

$0.40546510 \dots$