Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

$\int_{1}^{e^{2}} \frac{x^{3}}{x} + \frac{1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{e^{2}} \frac{x^{3}}{x} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.1

Factor $x$ out of $x^{3}$ .

$\int_{1}^{e^{2}} \frac{x \cdot x^{2}}{x} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\int_{1}^{e^{2}} \frac{x \cdot x^{2}}{x^{1}} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3.2.2

Factor $x$ out of $x^{1}$ .

$\int_{1}^{e^{2}} \frac{x \cdot x^{2}}{x \cdot 1} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3.2.3

Cancel the common factor.

$\int_{1}^{e^{2}} \frac{x \cdot x^{2}}{x \cdot 1} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3.2.4

Rewrite the expression.

$\int_{1}^{e^{2}} \frac{x^{2}}{1} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 3.2.5

Divide $x^{2}$ by $1$ .

$\int_{1}^{e^{2}} x^{2} d x + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{1}^{e^{2}} + \int_{1}^{e^{2}} \frac{1}{x} d x$

Step 5

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

$\frac{1}{3} x^{3}]_{1}^{e^{2}} + \ln (| x |)]_{1}^{e^{2}}$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{3} x^{3}]_{1}^{e^{2}}$ and $\ln (| x |)]_{1}^{e^{2}}$ .

$\frac{1}{3} x^{3} + \ln (| x |)]_{1}^{e^{2}}$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\frac{1}{3} x^{3} + \ln (| x |)$ at $e^{2}$ and at $1$ .

$(\frac{1}{3} {(e^{2})}^{3} + \ln (| e^{2} |)) - (\frac{1}{3} \cdot 1^{3} + \ln (| 1 |))$

Step 6.2.2

Simplify.

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Step 6.2.2.1

Multiply the exponents in ${(e^{2})}^{3}$ .

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Step 6.2.2.1.1

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

$\frac{1}{3} e^{2 \cdot 3} + \ln (| e^{2} |) - (\frac{1}{3} \cdot 1^{3} + \ln (| 1 |))$

Step 6.2.2.1.2

Multiply $2$ by $3$ .

$\frac{1}{3} e^{6} + \ln (| e^{2} |) - (\frac{1}{3} \cdot 1^{3} + \ln (| 1 |))$

Step 6.2.2.2

Combine $\frac{1}{3}$ and $e^{6}$ .

$\frac{e^{6}}{3} + \ln (| e^{2} |) - (\frac{1}{3} \cdot 1^{3} + \ln (| 1 |))$

Step 6.2.2.3

One to any power is one.

$\frac{e^{6}}{3} + \ln (| e^{2} |) - (\frac{1}{3} \cdot 1 + \ln (| 1 |))$

Step 6.2.2.4

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

$\frac{e^{6}}{3} + \ln (| e^{2} |) - (\frac{1}{3} + \ln (| 1 |))$

Step 6.2.2.5

To write $- (\frac{1}{3} + \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{e^{6}}{3} - (\frac{1}{3} + \ln (| 1 |)) \cdot \frac{3}{3} + \ln (| e^{2} |)$

Step 6.2.2.6

Combine $- (\frac{1}{3} + \ln (| 1 |))$ and $\frac{3}{3}$ .

$\frac{e^{6}}{3} + \frac{- (\frac{1}{3} + \ln (| 1 |)) \cdot 3}{3} + \ln (| e^{2} |)$

Step 6.2.2.7

Combine the numerators over the common denominator.

$\frac{e^{6} - (\frac{1}{3} + \ln (| 1 |)) \cdot 3}{3} + \ln (| e^{2} |)$

Step 6.2.2.8

Multiply $3$ by $- 1$ .

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + \ln (| e^{2} |)$

Step 7

Simplify each term.

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Step 7.1

$e^{2}$ is approximately $7.38905609$ which is positive so remove the absolute value

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + \ln (e^{2})$

Step 7.2

Expand $\ln (e^{2})$ by moving $2$ outside the logarithm.

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + 2 \ln (e)$

Step 7.3

The natural logarithm of $e$ is $1$ .

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + 2 \cdot 1$

Step 7.4

Multiply $2$ by $1$ .

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{6} - 3 (\frac{1}{3} + \ln (| 1 |))}{3} + 2$

Decimal Form:

$136.14293116 \dots$

Step 9