Calculus Examples

$\int_{2}^{x^{2}} \arctan (t) d t$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arctan (t)$ and $d v = 1$ .

$\arctan (t) t]_{2}^{x^{2}} - \int_{2}^{x^{2}} t \frac{1}{t^{2} + 1} d t$

Step 2

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

$\arctan (t) t]_{2}^{x^{2}} - \int_{2}^{x^{2}} \frac{t}{t^{2} + 1} d t$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $t^{2} + 1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

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

Step 3.1.4

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

$2 t + 0$

Step 3.1.5

Add $2 t$ and $0$ .

$2 t$

Step 3.2

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

$u_{lower} = 2^{2} + 1$

Step 3.3

Simplify.

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Step 3.3.1

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

$u_{lower} = 4 + 1$

Step 3.3.2

Add $4$ and $1$ .

$u_{lower} = 5$

Step 3.4

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

$u_{upper} = {(x^{2})}^{2} + 1$

Step 3.5

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.5.1

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

$u_{upper} = x^{2 \cdot 2} + 1$

Step 3.5.2

Multiply $2$ by $2$ .

$u_{upper} = x^{4} + 1$

Step 3.6

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

$u_{lower} = 5$

$u_{upper} = x^{4} + 1$

Step 3.7

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

$\arctan (t) t]_{2}^{x^{2}} - \int_{5}^{x^{4} + 1} \frac{1}{u} \cdot \frac{1}{2} d u$

Step 4

Simplify.

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Step 4.1

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

$\arctan (t) t]_{2}^{x^{2}} - \int_{5}^{x^{4} + 1} \frac{1}{u \cdot 2} d u$

Step 4.2

Move $2$ to the left of $u$ .

$\arctan (t) t]_{2}^{x^{2}} - \int_{5}^{x^{4} + 1} \frac{1}{2 u} d u$

Step 5

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

$\arctan (t) t]_{2}^{x^{2}} - (\frac{1}{2} \int_{5}^{x^{4} + 1} \frac{1}{u} d u)$

Step 6

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

$\arctan (t) t]_{2}^{x^{2}} - \frac{1}{2} \ln (| u |)]_{5}^{x^{4} + 1}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\arctan (t) t$ at $x^{2}$ and at $2$ .

$(\arctan (x^{2}) x^{2}) - \arctan (2) \cdot 2 - \frac{1}{2} \ln (| u |)]_{5}^{x^{4} + 1}$

Step 7.2

Evaluate $\ln (| u |)$ at $x^{4} + 1$ and at $5$ .

$(\arctan (x^{2}) x^{2}) - \arctan (2) \cdot 2 - \frac{1}{2} ((\ln (| x^{4} + 1 |)) - \ln (| 5 |))$

Step 7.3

Multiply $2$ by $- 1$ .

$\arctan (x^{2}) x^{2} - 2 \arctan (2) - \frac{1}{2} (\ln (| x^{4} + 1 |) - \ln (| 5 |))$

Step 8

Simplify.

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Step 8.1

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

$\arctan (x^{2}) x^{2} - 2 \arctan (2) - \frac{1}{2} \ln (\frac{| x^{4} + 1 |}{| 5 |})$

Step 8.2

Combine $\ln (\frac{| x^{4} + 1 |}{| 5 |})$ and $\frac{1}{2}$ .

$\arctan (x^{2}) x^{2} - 2 \arctan (2) - \frac{\ln (\frac{| x^{4} + 1 |}{| 5 |})}{2}$

Step 8.3

To write $\arctan (x^{2}) x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\arctan (x^{2}) x^{2} \cdot \frac{2}{2} - \frac{\ln (\frac{| x^{4} + 1 |}{| 5 |})}{2} - 2 \arctan (2)$

Step 8.4

Combine $\arctan (x^{2}) x^{2}$ and $\frac{2}{2}$ .

$\frac{\arctan (x^{2}) x^{2} \cdot 2}{2} - \frac{\ln (\frac{| x^{4} + 1 |}{| 5 |})}{2} - 2 \arctan (2)$

Step 8.5

Combine the numerators over the common denominator.

$\frac{\arctan (x^{2}) x^{2} \cdot 2 - \ln (\frac{| x^{4} + 1 |}{| 5 |})}{2} - 2 \arctan (2)$

Step 8.6

Move $2$ to the left of $\arctan (x^{2}) x^{2}$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (\frac{| x^{4} + 1 |}{| 5 |})}{2} - 2 \arctan (2)$

Step 9

Simplify terms.

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Step 9.1

Reorder terms.

$\frac{1}{2} (2 \arctan (x^{2}) x^{2} - \ln (\frac{1}{| 5 |} | x^{4} + 1 |)) - 2 \arctan (2)$

Step 9.2

Simplify each term.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{1}{2} (2 \arctan (x^{2}) x^{2} - \ln (\frac{1}{5} | x^{4} + 1 |)) - 2 \arctan (2)$

Step 9.2.1.2

Combine $\frac{1}{5}$ and $| x^{4} + 1 |$ .

$\frac{1}{2} (2 \arctan (x^{2}) x^{2} - \ln (\frac{| x^{4} + 1 |}{5})) - 2 \arctan (2)$

Step 9.2.2

Apply the distributive property.

$\frac{1}{2} (2 \arctan (x^{2}) x^{2}) + \frac{1}{2} (- \ln (\frac{| x^{4} + 1 |}{5})) - 2 \arctan (2)$

Step 9.2.3

Cancel the common factor of $2$ .

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Step 9.2.3.1

Factor $2$ out of $2 \arctan (x^{2}) x^{2}$ .

$\frac{1}{2} (2 (\arctan (x^{2}) x^{2})) + \frac{1}{2} (- \ln (\frac{| x^{4} + 1 |}{5})) - 2 \arctan (2)$

Step 9.2.3.2

Cancel the common factor.

$\frac{1}{2} (2 (\arctan (x^{2}) x^{2})) + \frac{1}{2} (- \ln (\frac{| x^{4} + 1 |}{5})) - 2 \arctan (2)$

Step 9.2.3.3

Rewrite the expression.

$\arctan (x^{2}) x^{2} + \frac{1}{2} (- \ln (\frac{| x^{4} + 1 |}{5})) - 2 \arctan (2)$

Step 9.2.4

Combine $\frac{1}{2}$ and $\ln (\frac{| x^{4} + 1 |}{5})$ .

$\arctan (x^{2}) x^{2} - \frac{\ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \arctan (2)$

Step 9.2.5

To write $\arctan (x^{2}) x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\arctan (x^{2}) x^{2} \cdot \frac{2}{2} - \frac{\ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \arctan (2)$

Step 9.2.6

Combine $\arctan (x^{2}) x^{2}$ and $\frac{2}{2}$ .

$\frac{\arctan (x^{2}) x^{2} \cdot 2}{2} - \frac{\ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \arctan (2)$

Step 9.2.7

Combine the numerators over the common denominator.

$\frac{\arctan (x^{2}) x^{2} \cdot 2 - \ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \arctan (2)$

Step 9.2.8

Move $2$ to the left of $\arctan (x^{2}) x^{2}$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \arctan (2)$

Step 9.2.9

Evaluate $\arctan (2)$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (\frac{| x^{4} + 1 |}{5})}{2} - 2 \cdot 1.10714871$

Step 9.2.10

Multiply $- 2$ by $1.10714871$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (\frac{| x^{4} + 1 |}{5})}{2} - 2.21429743$