Calculus Examples

$\int_{0}^{1} x^{19} + 19^{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{19} d x + \int_{0}^{1} 19^{x} d x$

Step 2

By the Power Rule, the integral of $x^{19}$ with respect to $x$ is $\frac{1}{20} x^{20}$ .

$\frac{1}{20} x^{20}]_{0}^{1} + \int_{0}^{1} 19^{x} d x$

Step 3

The integral of $19^{x}$ with respect to $x$ is $\frac{19^{x}}{\ln (19)}$ .

$\frac{1}{20} x^{20}]_{0}^{1} + \frac{19^{x}}{\ln (19)}]_{0}^{1}$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{20} x^{20}]_{0}^{1}$ and $\frac{19^{x}}{\ln (19)}]_{0}^{1}$ .

$\frac{1}{20} x^{20} + \frac{19^{x}}{\ln (19)}]_{0}^{1}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{1}{20} x^{20} + \frac{19^{x}}{\ln (19)}$ at $1$ and at $0$ .

$(\frac{1}{20} \cdot 1^{20} + \frac{19^{1}}{\ln (19)}) - (\frac{1}{20} \cdot 0^{20} + \frac{19^{0}}{\ln (19)})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

$\frac{1}{20} \cdot 1 + \frac{19^{1}}{\ln (19)} - (\frac{1}{20} \cdot 0^{20} + \frac{19^{0}}{\ln (19)})$

Step 4.2.2.2

Multiply $\frac{1}{20}$ by $1$ .

$\frac{1}{20} + \frac{19^{1}}{\ln (19)} - (\frac{1}{20} \cdot 0^{20} + \frac{19^{0}}{\ln (19)})$

Step 4.2.2.3

Evaluate the exponent.

$\frac{1}{20} + \frac{19}{\ln (19)} - (\frac{1}{20} \cdot 0^{20} + \frac{19^{0}}{\ln (19)})$

Step 4.2.2.4

Raising $0$ to any positive power yields $0$ .

$\frac{1}{20} + \frac{19}{\ln (19)} - (\frac{1}{20} \cdot 0 + \frac{19^{0}}{\ln (19)})$

Step 4.2.2.5

Multiply $\frac{1}{20}$ by $0$ .

$\frac{1}{20} + \frac{19}{\ln (19)} - (0 + \frac{19^{0}}{\ln (19)})$

Step 4.2.2.6

Anything raised to $0$ is $1$ .

$\frac{1}{20} + \frac{19}{\ln (19)} - (0 + \frac{1}{\ln (19)})$

Step 4.2.2.7

Add $0$ and $\frac{1}{\ln (19)}$ .

$\frac{1}{20} + \frac{19}{\ln (19)} - \frac{1}{\ln (19)}$

Step 4.2.2.8

Combine the numerators over the common denominator.

$\frac{1}{20} + \frac{19 - 1}{\ln (19)}$

Step 4.2.2.9

Subtract $1$ from $19$ .

$\frac{1}{20} + \frac{18}{\ln (19)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{20} + \frac{18}{\ln (19)}$

Decimal Form:

$6.16321889 \dots$

Step 6