Calculus Examples

$\int_{0}^{1} (2 x + 1) e^{x^{2} + x} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} + x$ .

$\frac{d}{d x} [x^{2} + x]$

Step 1.1.2

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

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [x]$

Step 1.1.4

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

$2 x + 1$

Step 1.2

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

$u_{lower} = 0^{2} + 0$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 0$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 2$

Step 1.7

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

$\int_{0}^{2} e^{u} d u$

Step 2

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

$e^{u}]_{0}^{2}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{u}$ at $2$ and at $0$ .

$(e^{2}) - e^{0}$

Step 3.2

Simplify.

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Step 3.2.1

Anything raised to $0$ is $1$ .

$e^{2} - 1 \cdot 1$

Step 3.2.2

Multiply $- 1$ by $1$ .

$e^{2} - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$e^{2} - 1$

Decimal Form:

$6.38905609 \dots$

Step 5