Calculus Examples

$\int_{0}^{1} 35 x^{4} e^{x^{5}} d x$

Step 1

Since $35$ is constant with respect to $x$ , move $35$ out of the integral.

$35 \int_{0}^{1} x^{4} e^{x^{5}} d x$

Step 2

Let $u_{2} = e^{x^{5}}$ . Then $d u_{2} = 5 x^{4} e^{x^{5}} d x$ , so $\frac{1}{5} d u_{2} = x^{4} e^{x^{5}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = e^{x^{5}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $e^{x^{5}}$ .

$\frac{d}{d x} [e^{x^{5}}]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{5}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{5}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{5}]$

Step 2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [x^{5}]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $x^{5}$ .

$e^{x^{5}} \frac{d}{d x} [x^{5}]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$e^{x^{5}} (5 x^{4})$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Reorder the factors of $e^{x^{5}} (5 x^{4})$ .

$5 e^{x^{5}} x^{4}$

Step 2.1.4.2

Reorder factors in $5 e^{x^{5}} x^{4}$ .

$5 x^{4} e^{x^{5}}$

Step 2.2

Substitute the lower limit in for $x$ in $u_{2} = e^{x^{5}}$ .

$u_{lower} = e^{0^{5}}$

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

Anything raised to $0$ is $1$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u_{2} = e^{x^{5}}$ .

$u_{upper} = e^{1^{5}}$

Step 2.5

Simplify.

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Step 2.5.1

One to any power is one.

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

Step 2.5.2

Simplify.

$u_{upper} = e$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = e$

Step 2.7

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

$35 \int_{1}^{e} \frac{1}{5} d u_{2}$

Step 3

Apply the constant rule.

$35 (\frac{1}{5} u_{2}]_{1}^{e})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{5}$ and $u_{2}$ .

$35 (\frac{u_{2}}{5}]_{1}^{e})$

Step 4.2

Evaluate $\frac{u_{2}}{5}$ at $e$ and at $1$ .

$35 (\frac{e}{5} - \frac{1}{5})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$35 \frac{e}{5} + 35 (- \frac{1}{5})$

Step 5.2

Cancel the common factor of $5$ .

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Step 5.2.1

Factor $5$ out of $35$ .

$5 (7) \frac{e}{5} + 35 (- \frac{1}{5})$

Step 5.2.2

Cancel the common factor.

$5 \cdot 7 \frac{e}{5} + 35 (- \frac{1}{5})$

Step 5.2.3

Rewrite the expression.

$7 e + 35 (- \frac{1}{5})$

Step 5.3

Cancel the common factor of $5$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$7 e + 35 (\frac{- 1}{5})$

Step 5.3.2

Factor $5$ out of $35$ .

$7 e + 5 (7) \frac{- 1}{5}$

Step 5.3.3

Cancel the common factor.

$7 e + 5 \cdot 7 \frac{- 1}{5}$

Step 5.3.4

Rewrite the expression.

$7 e + 7 \cdot - 1$

Step 5.4

Multiply $7$ by $- 1$ .

$7 e - 7$

Step 6

The result can be shown in multiple forms.

Exact Form:

$7 e - 7$

Decimal Form:

$12.02797279 \dots$

Step 7