Calculus Examples

$\int \frac{x^{2}}{e^{x^{3}}} d x$

Step 1

Let $u_{1} = x^{3}$ . Then $d u_{1} = 3 x^{2} d x$ , so $\frac{1}{3} d u_{1} = x^{2} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = x^{3}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $x^{3}$ .

$\frac{d}{d x} [x^{3}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2}$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{e^{u_{1}}} \cdot \frac{1}{3} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{e^{u_{1}}}$ by $\frac{1}{3}$ .

$\int \frac{1}{e^{u_{1}} \cdot 3} d u_{1}$

Step 2.2

Move $3$ to the left of $e^{u_{1}}$ .

$\int \frac{1}{3 e^{u_{1}}} d u_{1}$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{e^{u_{1}}} d u_{1}$

Step 4

Simplify the expression.

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Step 4.1

Negate the exponent of $e^{u_{1}}$ and move it out of the denominator.

$\frac{1}{3} \int 1 {(e^{u_{1}})}^{- 1} d u_{1}$

Step 4.2

Simplify.

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Step 4.2.1

Multiply the exponents in ${(e^{u_{1}})}^{- 1}$ .

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Step 4.2.1.1

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

$\frac{1}{3} \int 1 e^{u_{1} \cdot - 1} d u_{1}$

Step 4.2.1.2

Move $- 1$ to the left of $u_{1}$ .

$\frac{1}{3} \int 1 e^{- 1 \cdot u_{1}} d u_{1}$

Step 4.2.1.3

Rewrite $- 1 u_{1}$ as $- u_{1}$ .

$\frac{1}{3} \int 1 e^{- u_{1}} d u_{1}$

Step 4.2.2

Multiply $e^{- u_{1}}$ by $1$ .

$\frac{1}{3} \int e^{- u_{1}} d u_{1}$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $- u_{1}$ .

$\frac{d}{d u_{1}} [- u_{1}]$

Step 5.1.2

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- u_{1}$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [u_{1}]$ .

$- \frac{d}{d u_{1}} [u_{1}]$

Step 5.1.3

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

$- 1 \cdot 1$

Step 5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{3} \int - e^{u_{2}} d u_{2}$

Step 6

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

$\frac{1}{3} (- \int e^{u_{2}} d u_{2})$

Step 7

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

$\frac{1}{3} (- (e^{u_{2}} + C))$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{1}{3} (- e^{u_{2}}) + C$

Step 8.2

Combine $\frac{1}{3}$ and $e^{u_{2}}$ .

$- \frac{e^{u_{2}}}{3} + C$

Step 9

Substitute back in for each integration substitution variable.

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Step 9.1

Replace all occurrences of $u_{2}$ with $- u_{1}$ .

$- \frac{e^{- u_{1}}}{3} + C$

Step 9.2

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$- \frac{e^{- x^{3}}}{3} + C$

Step 10

Reorder terms.

$- \frac{1}{3} e^{- x^{3}} + C$