Calculus Examples

$\int_{\frac{1}{2}}^{1} \frac{1}{t^{2}} \cdot {(1 + \frac{1}{t})}^{3} d t$

Step 1

Combine $\frac{1}{t^{2}}$ and ${(1 + \frac{1}{t})}^{3}$ .

$\int_{\frac{1}{2}}^{1} \frac{{(1 + \frac{1}{t})}^{3}}{t^{2}} d t$

Step 2

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

Tap for more steps...

Step 2.1

Let $u = 1 + \frac{1}{t}$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 2.1.1

Differentiate $1 + \frac{1}{t}$ .

$\frac{d}{d t} [1 + \frac{1}{t}]$

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $1 + \frac{1}{t}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [\frac{1}{t}]$ .

$\frac{d}{d t} [1] + \frac{d}{d t} [\frac{1}{t}]$

Step 2.1.2.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [\frac{1}{t}]$

Step 2.1.3

Evaluate $\frac{d}{d t} [\frac{1}{t}]$ .

Tap for more steps...

Step 2.1.3.1

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

$0 + \frac{d}{d t} [t^{- 1}]$

Step 2.1.3.2

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

$0 - t^{- 2}$

Step 2.1.4

Simplify.

Tap for more steps...

Step 2.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - \frac{1}{t^{2}}$

Step 2.1.4.2

Subtract $\frac{1}{t^{2}}$ from $0$ .

$- \frac{1}{t^{2}}$

Step 2.2

Substitute the lower limit in for $t$ in $u = 1 + \frac{1}{t}$ .

$u_{lower} = 1 + \frac{1}{\frac{1}{2}}$

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 the numerator by the reciprocal of the denominator.

$u_{lower} = 1 + 1 \cdot 2$

Step 2.3.1.2

Multiply $2$ by $1$ .

$u_{lower} = 1 + 2$

Step 2.3.2

Add $1$ and $2$ .

$u_{lower} = 3$

Step 2.4

Substitute the upper limit in for $t$ in $u = 1 + \frac{1}{t}$ .

$u_{upper} = 1 + \frac{1}{1}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Divide $1$ by $1$ .

$u_{upper} = 1 + 1$

Step 2.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 2.6

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

$u_{lower} = 3$

$u_{upper} = 2$

Step 2.7

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

$\int_{3}^{2} - u^{3} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{3}^{2} u^{3} d u$

Step 4

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$- (\frac{1}{4} u^{4}]_{3}^{2})$

Step 5

Combine $\frac{1}{4}$ and $u^{4}$ .

$- (\frac{u^{4}}{4}]_{3}^{2})$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{u^{4}}{4}$ at $2$ and at $3$ .

$- ((\frac{2^{4}}{4}) - \frac{3^{4}}{4})$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Raise $2$ to the power of $4$ .

$- (\frac{16}{4} - \frac{3^{4}}{4})$

Step 6.2.2

Cancel the common factor of $16$ and $4$ .

Tap for more steps...

Step 6.2.2.1

Factor $4$ out of $16$ .

$- (\frac{4 \cdot 4}{4} - \frac{3^{4}}{4})$

Step 6.2.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.2.1

Factor $4$ out of $4$ .

$- (\frac{4 \cdot 4}{4 (1)} - \frac{3^{4}}{4})$

Step 6.2.2.2.2

Cancel the common factor.

$- (\frac{4 \cdot 4}{4 \cdot 1} - \frac{3^{4}}{4})$

Step 6.2.2.2.3

Rewrite the expression.

$- (\frac{4}{1} - \frac{3^{4}}{4})$

Step 6.2.2.2.4

Divide $4$ by $1$ .

$- (4 - \frac{3^{4}}{4})$

Step 6.2.3

Raise $3$ to the power of $4$ .

$- (4 - \frac{81}{4})$

Step 6.2.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- (4 \cdot \frac{4}{4} - \frac{81}{4})$

Step 6.2.5

Combine $4$ and $\frac{4}{4}$ .

$- (\frac{4 \cdot 4}{4} - \frac{81}{4})$

Step 6.2.6

Combine the numerators over the common denominator.

$- \frac{4 \cdot 4 - 81}{4}$

Step 6.2.7

Simplify the numerator.

Tap for more steps...

Step 6.2.7.1

Multiply $4$ by $4$ .

$- \frac{16 - 81}{4}$

Step 6.2.7.2

Subtract $81$ from $16$ .

$- \frac{- 65}{4}$

Step 6.2.8

Move the negative in front of the fraction.

$- - \frac{65}{4}$

Step 6.2.9

Multiply $- 1$ by $- 1$ .

$1 (\frac{65}{4})$

Step 6.2.10

Multiply $\frac{65}{4}$ by $1$ .

$\frac{65}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{65}{4}$

Decimal Form:

$16.25$

Mixed Number Form:

$16 \frac{1}{4}$

Step 8