Calculus Examples

$\int_{0}^{45} \frac{1}{\sqrt{t + 4}} d t$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $t + 4$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$u_{lower} = 0 + 4$

Step 1.3

Add $0$ and $4$ .

$u_{lower} = 4$

Step 1.4

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

$u_{upper} = 45 + 4$

Step 1.5

Add $45$ and $4$ .

$u_{upper} = 49$

Step 1.6

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

$u_{lower} = 4$

$u_{upper} = 49$

Step 1.7

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

$\int_{4}^{49} \frac{1}{\sqrt{u}} d u$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int_{4}^{49} \frac{1}{u^{\frac{1}{2}}} d u$

Step 2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{4}^{49} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.1

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

$\int_{4}^{49} u^{\frac{1}{2} \cdot - 1} d u$

Step 2.3.2

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

$\int_{4}^{49} u^{\frac{- 1}{2}} d u$

Step 2.3.3

Move the negative in front of the fraction.

$\int_{4}^{49} u^{- \frac{1}{2}} d u$

Step 3

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$2 u^{\frac{1}{2}}]_{4}^{49}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $2 u^{\frac{1}{2}}$ at $49$ and at $4$ .

$(2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Rewrite $49$ as $7^{2}$ .

$2 \cdot {(7^{2})}^{\frac{1}{2}} - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.2

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

$2 \cdot 7^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$2 \cdot 7^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.3.2

Rewrite the expression.

$2 \cdot 7^{1} - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.4

Evaluate the exponent.

$2 \cdot 7 - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.5

Multiply $2$ by $7$ .

$14 - 2 \cdot 4^{\frac{1}{2}}$

Step 4.2.6

Rewrite $4$ as $2^{2}$ .

$14 - 2 \cdot {(2^{2})}^{\frac{1}{2}}$

Step 4.2.7

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

$14 - 2 \cdot 2^{2 (\frac{1}{2})}$

Step 4.2.8

Cancel the common factor of $2$ .

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Step 4.2.8.1

Cancel the common factor.

$14 - 2 \cdot 2^{2 (\frac{1}{2})}$

Step 4.2.8.2

Rewrite the expression.

$14 - 2 \cdot 2^{1}$

Step 4.2.9

Evaluate the exponent.

$14 - 2 \cdot 2$

Step 4.2.10

Multiply $- 2$ by $2$ .

$14 - 4$

Step 4.2.11

Subtract $4$ from $14$ .

$10$

Step 5