Calculus Examples

$\frac{d y}{d x} = - 3 x^{2} e^{- x^{3}}$

Step 1

Rewrite the equation.

$d y = - 3 x^{2} e^{- x^{3}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 3 x^{2} e^{- x^{3}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 3 x^{2} e^{- x^{3}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$y + C_{1} = - 3 \int x^{2} e^{- x^{3}} d x$

Step 2.3.2

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

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Step 2.3.2.1

Let $u_{2} = e^{- x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $e^{- x^{3}}$ .

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

Step 2.3.2.1.2

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 $g (x) = - x^{3}$ .

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Step 2.3.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{3}]$

Step 2.3.2.1.2.2

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

$e^{u_{1}} \frac{d}{d x} [- x^{3}]$

Step 2.3.2.1.2.3

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

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

Step 2.3.2.1.3

Differentiate.

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Step 2.3.2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$e^{- x^{3}} (- \frac{d}{d x} [x^{3}])$

Step 2.3.2.1.3.2

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

$e^{- x^{3}} (- (3 x^{2}))$

Step 2.3.2.1.3.3

Multiply $3$ by $- 1$ .

$e^{- x^{3}} (- 3 x^{2})$

Step 2.3.2.1.4

Simplify.

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Step 2.3.2.1.4.1

Reorder the factors of $e^{- x^{3}} (- 3 x^{2})$ .

$- 3 e^{- x^{3}} x^{2}$

Step 2.3.2.1.4.2

Reorder factors in $- 3 e^{- x^{3}} x^{2}$ .

$- 3 x^{2} e^{- x^{3}}$

Step 2.3.2.2

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

$y + C_{1} = - 3 \int \frac{1}{- 3} d u_{2}$

Step 2.3.3

Move the negative in front of the fraction.

$y + C_{1} = - 3 \int - \frac{1}{3} d u_{2}$

Step 2.3.4

Apply the constant rule.

$y + C_{1} = - 3 (- \frac{1}{3} u_{2} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Simplify.

$y + C_{1} = - 3 (- \frac{1}{3} u_{2}) + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

$y + C_{1} = - 3 (- \frac{u_{2}}{3}) + C_{2}$

Step 2.3.5.2.2

Multiply $- 1$ by $- 3$ .

$y + C_{1} = 3 \frac{u_{2}}{3} + C_{2}$

Step 2.3.5.2.3

Combine $3$ and $\frac{u_{2}}{3}$ .

$y + C_{1} = \frac{3 u_{2}}{3} + C_{2}$

Step 2.3.5.2.4

Cancel the common factor of $3$ .

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Step 2.3.5.2.4.1

Cancel the common factor.

$y + C_{1} = \frac{3 u_{2}}{3} + C_{2}$

Step 2.3.5.2.4.2

Divide $u_{2}$ by $1$ .

$y + C_{1} = u_{2} + C_{2}$

Step 2.3.5.3

Replace all occurrences of $u_{2}$ with $e^{- x^{3}}$ .

$y + C_{1} = e^{- x^{3}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = e^{- x^{3}} + K$