Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$\frac{1}{e + 1} x^{e + 1}]_{0}^{1} + e^{x}]_{0}^{1}$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{e + 1} x^{e + 1}]_{0}^{1}$ and $e^{x}]_{0}^{1}$ .

$\frac{1}{e + 1} x^{e + 1} + e^{x}]_{0}^{1}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{1}{e + 1} x^{e + 1} + e^{x}$ at $1$ and at $0$ .

$(\frac{1}{e + 1} \cdot 1^{e + 1} + e^{1}) - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

$\frac{1}{e + 1} \cdot 1 + e^{1} - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2.2

Multiply $\frac{1}{e + 1}$ by $1$ .

$\frac{1}{e + 1} + e^{1} - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2.3

Simplify.

$\frac{1}{e + 1} + e - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2.4

To write $e$ as a fraction with a common denominator, multiply by $\frac{e + 1}{e + 1}$ .

$\frac{1}{e + 1} + \frac{e (e + 1)}{e + 1} - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2.5

Combine the numerators over the common denominator.

$\frac{1 + e (e + 1)}{e + 1} - (\frac{1}{e + 1} \cdot 0^{e + 1} + e^{0})$

Step 4.2.2.6

Combine $\frac{1}{e + 1}$ and $0^{e + 1}$ .

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

Step 4.2.2.7

Anything raised to $0$ is $1$ .

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

Step 4.2.2.8

Write $1$ as a fraction with a common denominator.

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

Step 4.2.2.9

Combine the numerators over the common denominator.

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

Step 4.2.2.10

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{1 + e e + e \cdot 1 - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.2

Multiply $e e$ .

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Step 5.2.1

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

$\frac{1 + e^{1} e + e \cdot 1 - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.2.2

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

$\frac{1 + e^{1} e^{1} + e \cdot 1 - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.2.3

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

$\frac{1 + e^{1 + 1} + e \cdot 1 - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.2.4

Add $1$ and $1$ .

$\frac{1 + e^{2} + e \cdot 1 - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.3

Multiply $e$ by $1$ .

$\frac{1 + e^{2} + e - (0^{e + 1} + e + 1)}{e + 1}$

Step 5.4

Apply the distributive property.

$\frac{1 + e^{2} + e - 0^{e + 1} - e - 1 \cdot 1}{e + 1}$

Step 5.5

Multiply $- 1$ by $1$ .

$\frac{1 + e^{2} + e - 0^{e + 1} - e - 1}{e + 1}$

Step 5.6

Subtract $1$ from $1$ .

$\frac{0 + e^{2} + e - 0^{e + 1} - e}{e + 1}$

Step 5.7

Add $0$ and $e^{2}$ .

$\frac{e^{2} + e - 0^{e + 1} - e}{e + 1}$

Step 5.8

Subtract $e$ from $e$ .

$\frac{e^{2} + 0 - 0^{e + 1}}{e + 1}$

Step 5.9

Add $e^{2}$ and $0$ .

$\frac{e^{2} - 0^{e + 1}}{e + 1}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{2} - 0^{e + 1}}{e + 1}$

Decimal Form:

$1.98722324 \dots$

Step 7