Calculus Examples

$\int_{0}^{1} x \cdot 5^{x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = 5^{x}$ .

$x (\frac{1}{\ln (5)} \cdot 5^{x})]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (5)} \cdot 5^{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{\ln (5)}$ and $5^{x}$ .

$x \frac{5^{x}}{\ln (5)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (5)} \cdot 5^{x} d x$

Step 2.2

Combine $x$ and $\frac{5^{x}}{\ln (5)}$ .

$\frac{x \cdot 5^{x}}{\ln (5)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (5)} \cdot 5^{x} d x$

Step 3

Since $\frac{1}{\ln (5)}$ is constant with respect to $x$ , move $\frac{1}{\ln (5)}$ out of the integral.

$\frac{x \cdot 5^{x}}{\ln (5)}]_{0}^{1} - (\frac{1}{\ln (5)} \int_{0}^{1} 5^{x} d x)$

Step 4

The integral of $5^{x}$ with respect to $x$ is $\frac{5^{x}}{\ln (5)}$ .

$\frac{x \cdot 5^{x}}{\ln (5)}]_{0}^{1} - \frac{1}{\ln (5)} \frac{5^{x}}{\ln (5)}]_{0}^{1}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{5^{x}}{\ln (5)}]_{0}^{1}$ and $\frac{1}{\ln (5)}$ .

$\frac{x \cdot 5^{x}}{\ln (5)}]_{0}^{1} - \frac{\frac{5^{x}}{\ln (5)}]_{0}^{1}}{\ln (5)}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{x \cdot 5^{x}}{\ln (5)}$ at $1$ and at $0$ .

$(\frac{1 \cdot 5^{1}}{\ln (5)}) - \frac{0 \cdot 5^{0}}{\ln (5)} - \frac{\frac{5^{x}}{\ln (5)}]_{0}^{1}}{\ln (5)}$

Step 5.2.2

Evaluate $\frac{5^{x}}{\ln (5)}$ at $1$ and at $0$ .

$(\frac{1 \cdot 5^{1}}{\ln (5)}) - \frac{0 \cdot 5^{0}}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Evaluate the exponent.

$\frac{1 \cdot 5}{\ln (5)} - \frac{0 \cdot 5^{0}}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.2

Multiply $5$ by $1$ .

$\frac{5}{\ln (5)} - \frac{0 \cdot 5^{0}}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.3

Anything raised to $0$ is $1$ .

$\frac{5}{\ln (5)} - \frac{0 \cdot 1}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.4

Multiply $0$ by $1$ .

$\frac{5}{\ln (5)} - \frac{0}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.5

Combine the numerators over the common denominator.

$\frac{5 + 0}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.6

Add $5$ and $0$ .

$\frac{5}{\ln (5)} - \frac{(\frac{5^{1}}{\ln (5)}) - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.7

Evaluate the exponent.

$\frac{5}{\ln (5)} - \frac{\frac{5}{\ln (5)} - \frac{5^{0}}{\ln (5)}}{\ln (5)}$

Step 5.2.3.8

Anything raised to $0$ is $1$ .

$\frac{5}{\ln (5)} - \frac{\frac{5}{\ln (5)} - \frac{1}{\ln (5)}}{\ln (5)}$

Step 5.2.3.9

Combine the numerators over the common denominator.

$\frac{5}{\ln (5)} - \frac{\frac{5 - 1}{\ln (5)}}{\ln (5)}$

Step 5.2.3.10

Subtract $1$ from $5$ .

$\frac{5}{\ln (5)} - \frac{\frac{4}{\ln (5)}}{\ln (5)}$

Step 5.2.3.11

Rewrite $\frac{\frac{4}{\ln (5)}}{\ln (5)}$ as a product.

$\frac{5}{\ln (5)} - (\frac{4}{\ln (5)} \cdot \frac{1}{\ln (5)})$

Step 5.2.3.12

Multiply $\frac{4}{\ln (5)}$ by $\frac{1}{\ln (5)}$ .

$\frac{5}{\ln (5)} - \frac{4}{\ln (5) \ln (5)}$

Step 5.2.3.13

Raise $\ln (5)$ to the power of $1$ .

$\frac{5}{\ln (5)} - \frac{4}{\ln^{1} (5) \ln (5)}$

Step 5.2.3.14

Raise $\ln (5)$ to the power of $1$ .

$\frac{5}{\ln (5)} - \frac{4}{\ln^{1} (5) \ln^{1} (5)}$

Step 5.2.3.15

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

$\frac{5}{\ln (5)} - \frac{4}{{\ln (5)}^{1 + 1}}$

Step 5.2.3.16

Add $1$ and $1$ .

$\frac{5}{\ln (5)} - \frac{4}{\ln^{2} (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{\ln (5)} - \frac{4}{\ln^{2} (5)}$

Decimal Form:

$1.56244626 \dots$

Step 7