Calculus Examples

$\int_{0}^{1} 4 x^{3} e^{x^{4}} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{0}^{1} x^{3} e^{x^{4}} d x$

Step 2

Let $u_{1} = x^{2}$ . Then $d u_{1} = 2 x d x$ , so $\frac{1}{2} d u_{1} = x d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

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

$2 x$

Step 2.2

Substitute the lower limit in for $x$ in $u_{1} = x^{2}$ .

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

Step 2.3

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

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u_{1} = x^{2}$ .

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

Step 2.5

One to any power is one.

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 2.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$4 \int_{0}^{1} u_{1} e^{{\sqrt{u_{1}}}^{4}} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

$4 \int_{0}^{1} u_{1} e^{{({u_{1}}^{\frac{1}{2}})}^{4}} \frac{1}{2} d u_{1}$

Step 3.1.2

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

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{1}$

Step 3.1.3

Combine $\frac{1}{2}$ and $4$ .

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{4}{2}}} \frac{1}{2} d u_{1}$

Step 3.1.4

Cancel the common factor of $4$ and $2$ .

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Step 3.1.4.1

Factor $2$ out of $4$ .

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{2 \cdot 2}{2}}} \frac{1}{2} d u_{1}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{2 \cdot 2}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.2

Cancel the common factor.

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.3

Rewrite the expression.

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{\frac{2}{1}}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.4

Divide $2$ by $1$ .

$4 \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} \frac{1}{2} d u_{1}$

Step 3.2

Combine $\frac{1}{2}$ and $u_{1}$ .

$4 \int_{0}^{1} \frac{u_{1}}{2} e^{{u_{1}}^{2}} d u_{1}$

Step 3.3

Combine $\frac{u_{1}}{2}$ and $e^{{u_{1}}^{2}}$ .

$4 \int_{0}^{1} \frac{u_{1} e^{{u_{1}}^{2}}}{2} d u_{1}$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$4 (\frac{1}{2} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 5.2

Cancel the common factor of $4$ and $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 5.2.2.3

Rewrite the expression.

$\frac{2}{1} \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 5.2.2.4

Divide $2$ by $1$ .

$2 \int_{0}^{1} u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 6

Let $u_{2} = {u_{1}}^{2}$ . Then $d u_{2} = 2 u_{1} d u_{1}$ , so $\frac{1}{2} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate ${u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 6.1.2

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

$2 u_{1}$

Step 6.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

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

Step 6.3

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

$u_{lower} = 0$

Step 6.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

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

Step 6.5

One to any power is one.

$u_{upper} = 1$

Step 6.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 6.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$2 \int_{0}^{1} e^{u_{2}} \frac{1}{2} d u_{2}$

Step 7

Combine $e^{u_{2}}$ and $\frac{1}{2}$ .

$2 \int_{0}^{1} \frac{e^{u_{2}}}{2} d u_{2}$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int_{0}^{1} e^{u_{2}} d u_{2})$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int_{0}^{1} e^{u_{2}} d u_{2}$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

$\frac{2}{2} \int_{0}^{1} e^{u_{2}} d u_{2}$

Step 9.2.2

Rewrite the expression.

$1 \int_{0}^{1} e^{u_{2}} d u_{2}$

Step 9.3

Multiply $\int_{0}^{1} e^{u_{2}} d u_{2}$ by $1$ .

$\int_{0}^{1} e^{u_{2}} d u_{2}$

Step 10

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$e^{u_{2}}]_{0}^{1}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $e^{u_{2}}$ at $1$ and at $0$ .

$(e^{1}) - e^{0}$

Step 11.2

Simplify.

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Step 11.2.1

Simplify.

$e - e^{0}$

Step 11.2.2

Anything raised to $0$ is $1$ .

$e - 1 \cdot 1$

Step 11.2.3

Multiply $- 1$ by $1$ .

$e - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$e - 1$

Decimal Form:

$1.71828182 \dots$

Step 13