Calculus Examples

$\int_{4}^{7} \sqrt{4 t^{2} + 4 t + 1} d t$

Step 1

Let $u = \sqrt{4 t^{2} + 4 t + 1}$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \sqrt{4 t^{2} + 4 t + 1}$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $\sqrt{4 t^{2} + 4 t + 1}$ .

$\frac{d}{d t} [\sqrt{4 t^{2} + 4 t + 1}]$

Step 1.1.2

Rewrite $\frac{d}{d t} [\sqrt{4 t^{2} + 4 t + 1}]$ as $\frac{d}{d t} [2 t + 1]$ .

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Step 1.1.2.1

Factor using the perfect square rule.

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Step 1.1.2.1.1

Rewrite $4 t^{2}$ as ${(2 t)}^{2}$ .

$\frac{d}{d t} [\sqrt{{(2 t)}^{2} + 4 t + 1}]$

Step 1.1.2.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{d}{d t} [\sqrt{{(2 t)}^{2} + 4 t + 1^{2}}]$

Step 1.1.2.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 t = 2 \cdot (2 t) \cdot 1$

Step 1.1.2.1.4

Rewrite the polynomial.

$\frac{d}{d t} [\sqrt{{(2 t)}^{2} + 2 \cdot (2 t) \cdot 1 + 1^{2}}]$

Step 1.1.2.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 t$ and $b = 1$ .

$\frac{d}{d t} [\sqrt{{(2 t + 1)}^{2}}]$

Step 1.1.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{d}{d t} [2 t + 1]$

Step 1.1.3

By the Sum Rule, the derivative of $2 t + 1$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [1]$ .

$\frac{d}{d t} [2 t] + \frac{d}{d t} [1]$

Step 1.1.4

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

$2 \frac{d}{d t} [t] + \frac{d}{d t} [1]$

Step 1.1.5

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

$2 \cdot 1 + \frac{d}{d t} [1]$

Step 1.1.6

Multiply $2$ by $1$ .

$2 + \frac{d}{d t} [1]$

Step 1.1.7

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

$2 + 0$

Step 1.1.8

Add $2$ and $0$ .

$2$

Step 1.2

Substitute the lower limit in for $t$ in $u = \sqrt{4 t^{2} + 4 t + 1}$ .

$u_{lower} = \sqrt{4 \cdot 4^{2} + 4 \cdot 4 + 1}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $4$ by $4^{2}$ by adding the exponents.

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Step 1.3.1.1

Multiply $4$ by $4^{2}$ .

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Step 1.3.1.1.1

Raise $4$ to the power of $1$ .

$u_{lower} = \sqrt{4^{1} \cdot 4^{2} + 4 \cdot 4 + 1}$

Step 1.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u_{lower} = \sqrt{4^{1 + 2} + 4 \cdot 4 + 1}$

Step 1.3.1.2

Add $1$ and $2$ .

$u_{lower} = \sqrt{4^{3} + 4 \cdot 4 + 1}$

Step 1.3.2

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

$u_{lower} = \sqrt{64 + 4 \cdot 4 + 1}$

Step 1.3.3

Multiply $4$ by $4$ .

$u_{lower} = \sqrt{64 + 16 + 1}$

Step 1.3.4

Add $64$ and $16$ .

$u_{lower} = \sqrt{80 + 1}$

Step 1.3.5

Add $80$ and $1$ .

$u_{lower} = \sqrt{81}$

Step 1.3.6

Rewrite $81$ as $9^{2}$ .

$u_{lower} = \sqrt{9^{2}}$

Step 1.3.7

Pull terms out from under the radical, assuming positive real numbers.

$u_{lower} = 9$

Step 1.4

Substitute the upper limit in for $t$ in $u = \sqrt{4 t^{2} + 4 t + 1}$ .

$u_{upper} = \sqrt{4 \cdot 7^{2} + 4 \cdot 7 + 1}$

Step 1.5

Simplify.

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Step 1.5.1

Raise $7$ to the power of $2$ .

$u_{upper} = \sqrt{4 \cdot 49 + 4 \cdot 7 + 1}$

Step 1.5.2

Multiply $4$ by $49$ .

$u_{upper} = \sqrt{196 + 4 \cdot 7 + 1}$

Step 1.5.3

Multiply $4$ by $7$ .

$u_{upper} = \sqrt{196 + 28 + 1}$

Step 1.5.4

Add $196$ and $28$ .

$u_{upper} = \sqrt{224 + 1}$

Step 1.5.5

Add $224$ and $1$ .

$u_{upper} = \sqrt{225}$

Step 1.5.6

Rewrite $225$ as $15^{2}$ .

$u_{upper} = \sqrt{15^{2}}$

Step 1.5.7

Pull terms out from under the radical, assuming positive real numbers.

$u_{upper} = 15$

Step 1.6

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

$u_{lower} = 9$

$u_{upper} = 15$

Step 1.7

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

$\int_{9}^{15} u \frac{1}{2} d u$

Step 2

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

$\int_{9}^{15} \frac{u}{2} d u$

Step 3

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

$\frac{1}{2} \int_{9}^{15} u d u$

Step 4

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

$\frac{1}{2} \frac{1}{2} u^{2}]_{9}^{15}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{2} u^{2}$ at $15$ and at $9$ .

$\frac{1}{2} ((\frac{1}{2} \cdot 15^{2}) - \frac{1}{2} \cdot 9^{2})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $15$ to the power of $2$ .

$\frac{1}{2} (\frac{1}{2} \cdot 225 - \frac{1}{2} \cdot 9^{2})$

Step 5.2.2

Combine $\frac{1}{2}$ and $225$ .

$\frac{1}{2} (\frac{225}{2} - \frac{1}{2} \cdot 9^{2})$

Step 5.2.3

Raise $9$ to the power of $2$ .

$\frac{1}{2} (\frac{225}{2} - \frac{1}{2} \cdot 81)$

Step 5.2.4

Multiply $81$ by $- 1$ .

$\frac{1}{2} (\frac{225}{2} - 81 (\frac{1}{2}))$

Step 5.2.5

Combine $- 81$ and $\frac{1}{2}$ .

$\frac{1}{2} (\frac{225}{2} + \frac{- 81}{2})$

Step 5.2.6

Move the negative in front of the fraction.

$\frac{1}{2} (\frac{225}{2} - \frac{81}{2})$

Step 5.2.7

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{225 - 81}{2}$

Step 5.2.8

Subtract $81$ from $225$ .

$\frac{1}{2} \cdot \frac{144}{2}$

Step 5.2.9

Cancel the common factor of $144$ and $2$ .

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Step 5.2.9.1

Factor $2$ out of $144$ .

$\frac{1}{2} \cdot \frac{2 \cdot 72}{2}$

Step 5.2.9.2

Cancel the common factors.

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Step 5.2.9.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \cdot \frac{2 \cdot 72}{2 (1)}$

Step 5.2.9.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 72}{2 \cdot 1}$

Step 5.2.9.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{72}{1}$

Step 5.2.9.2.4

Divide $72$ by $1$ .

$\frac{1}{2} \cdot 72$

Step 5.2.10

Combine $\frac{1}{2}$ and $72$ .

$\frac{72}{2}$

Step 5.2.11

Cancel the common factor of $72$ and $2$ .

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Step 5.2.11.1

Factor $2$ out of $72$ .

$\frac{2 \cdot 36}{2}$

Step 5.2.11.2

Cancel the common factors.

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Step 5.2.11.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 36}{2 (1)}$

Step 5.2.11.2.2

Cancel the common factor.

$\frac{2 \cdot 36}{2 \cdot 1}$

Step 5.2.11.2.3

Rewrite the expression.

$\frac{36}{1}$

Step 5.2.11.2.4

Divide $36$ by $1$ .

$36$

Step 6