Calculus Examples

$\int_{0}^{\infty} \frac{5}{{(x + 4)}^{4}} d x$

Step 1

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{0}^{t} \frac{5}{{(x + 4)}^{4}} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$lim_{t \to \infty} 5 \int_{0}^{t} \frac{1}{{(x + 4)}^{4}} d x$

Step 3

Let $u = x + 4$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x + 4$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x + 4$ .

$\frac{d}{d x} [x + 4]$

Step 3.1.2

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 3.1.3

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

$1 + \frac{d}{d x} [4]$

Step 3.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

Substitute the lower limit in for $x$ in $u = x + 4$ .

$u_{lower} = 0 + 4$

Step 3.3

Add $0$ and $4$ .

$u_{lower} = 4$

Step 3.4

Substitute the upper limit in for $x$ in $u = x + 4$ .

$u_{upper} = t + 4$

Step 3.5

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

$u_{lower} = 4$

$u_{upper} = t + 4$

Step 3.6

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

$lim_{t \to \infty} 5 \int_{4}^{t + 4} \frac{1}{u^{4}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{4}$ out of the denominator by raising it to the $- 1$ power.

$lim_{t \to \infty} 5 \int_{4}^{t + 4} {(u^{4})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{4})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$lim_{t \to \infty} 5 \int_{4}^{t + 4} u^{4 \cdot - 1} d u$

Step 4.2.2

Multiply $4$ by $- 1$ .

$lim_{t \to \infty} 5 \int_{4}^{t + 4} u^{- 4} d u$

Step 5

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

$lim_{t \to \infty} 5 (- \frac{1}{3} u^{- 3}]_{4}^{t + 4})$

Step 6

Simplify.

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Step 6.1

Combine $u^{- 3}$ and $\frac{1}{3}$ .

$lim_{t \to \infty} 5 (- \frac{u^{- 3}}{3}]_{4}^{t + 4})$

Step 6.2

Move $u^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to \infty} 5 (- \frac{1}{3 u^{3}}]_{4}^{t + 4})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $- \frac{1}{3 u^{3}}$ at $t + 4$ and at $4$ .

$lim_{t \to \infty} 5 ((- \frac{1}{3 {(t + 4)}^{3}}) + \frac{1}{3 \cdot 4^{3}})$

Step 7.2

Simplify.

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Step 7.2.1

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

$lim_{t \to \infty} 5 (- \frac{1}{3 {(t + 4)}^{3}} + \frac{1}{3 \cdot 64})$

Step 7.2.2

Multiply $3$ by $64$ .

$lim_{t \to \infty} 5 (- \frac{1}{3 {(t + 4)}^{3}} + \frac{1}{192})$

Step 8

Evaluate the limit.

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Step 8.1

Evaluate the limit.

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Step 8.1.1

Move the term $5$ outside of the limit because it is constant with respect to $t$ .

$5 lim_{t \to \infty} - \frac{1}{3 {(t + 4)}^{3}} + \frac{1}{192}$

Step 8.1.2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\infty$ .

$5 (- lim_{t \to \infty} \frac{1}{3 {(t + 4)}^{3}} + lim_{t \to \infty} \frac{1}{192})$

Step 8.1.3

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $t$ .

$5 (- \frac{1}{3} lim_{t \to \infty} \frac{1}{{(t + 4)}^{3}} + lim_{t \to \infty} \frac{1}{192})$

Step 8.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{{(t + 4)}^{3}}$ approaches $0$ .

$5 (- \frac{1}{3} \cdot 0 + lim_{t \to \infty} \frac{1}{192})$

Step 8.3

Evaluate the limit.

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Step 8.3.1

Evaluate the limit of $\frac{1}{192}$ which is constant as $t$ approaches $\infty$ .

$5 (- \frac{1}{3} \cdot 0 + \frac{1}{192})$

Step 8.3.2

Simplify the answer.

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Step 8.3.2.1

Multiply $- \frac{1}{3} \cdot 0$ .

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Step 8.3.2.1.1

Multiply $0$ by $- 1$ .

$5 (0 (\frac{1}{3}) + \frac{1}{192})$

Step 8.3.2.1.2

Multiply $0$ by $\frac{1}{3}$ .

$5 (0 + \frac{1}{192})$

Step 8.3.2.2

Add $0$ and $\frac{1}{192}$ .

$5 (\frac{1}{192})$

Step 8.3.2.3

Combine $5$ and $\frac{1}{192}$ .

$\frac{5}{192}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{192}$

Decimal Form:

$0.026041\overline{6}$