Calculus Examples

$\frac{d y}{d x} = - 9 x^{8} e^{- x^{9}}$ , $y (0) = 2$

Step 1

Rewrite the equation.

$d y = - 9 x^{8} e^{- x^{9}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 9 x^{8} e^{- x^{9}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 9 x^{8} e^{- x^{9}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$y + C_{1} = - 9 \int x^{8} e^{- x^{9}} d x$

Step 2.3.2

Let $u_{2} = e^{- x^{9}}$ . Then $d u_{2} = - 9 x^{8} e^{- x^{9}} d x$ , so $- \frac{1}{9} d u_{2} = x^{8} e^{- x^{9}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.2.1

Let $u_{2} = e^{- x^{9}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $e^{- x^{9}}$ .

$\frac{d}{d x} [e^{- x^{9}}]$

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 $g (x) = - x^{9}$ .

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Step 2.3.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- x^{9}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{9}]$

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 $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{9}]$

Step 2.3.2.1.2.3

Replace all occurrences of $u_{1}$ with $- x^{9}$ .

$e^{- x^{9}} \frac{d}{d x} [- x^{9}]$

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 $- x^{9}$ with respect to $x$ is $- \frac{d}{d x} [x^{9}]$ .

$e^{- x^{9}} (- \frac{d}{d x} [x^{9}])$

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 $n = 9$ .

$e^{- x^{9}} (- (9 x^{8}))$

Step 2.3.2.1.3.3

Multiply $9$ by $- 1$ .

$e^{- x^{9}} (- 9 x^{8})$

Step 2.3.2.1.4

Simplify.

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Step 2.3.2.1.4.1

Reorder the factors of $e^{- x^{9}} (- 9 x^{8})$ .

$- 9 e^{- x^{9}} x^{8}$

Step 2.3.2.1.4.2

Reorder factors in $- 9 e^{- x^{9}} x^{8}$ .

$- 9 x^{8} e^{- x^{9}}$

Step 2.3.2.2

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

$y + C_{1} = - 9 \int \frac{1}{- 9} d u_{2}$

Step 2.3.3

Move the negative in front of the fraction.

$y + C_{1} = - 9 \int - \frac{1}{9} d u_{2}$

Step 2.3.4

Apply the constant rule.

$y + C_{1} = - 9 (- \frac{1}{9} u_{2} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Simplify.

$y + C_{1} = - 9 (- \frac{1}{9} u_{2}) + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $u_{2}$ and $\frac{1}{9}$ .

$y + C_{1} = - 9 (- \frac{u_{2}}{9}) + C_{2}$

Step 2.3.5.2.2

Multiply $- 1$ by $- 9$ .

$y + C_{1} = 9 \frac{u_{2}}{9} + C_{2}$

Step 2.3.5.2.3

Combine $9$ and $\frac{u_{2}}{9}$ .

$y + C_{1} = \frac{9 u_{2}}{9} + C_{2}$

Step 2.3.5.2.4

Cancel the common factor of $9$ .

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Step 2.3.5.2.4.1

Cancel the common factor.

$y + C_{1} = \frac{9 u_{2}}{9} + C_{2}$

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 $e^{- x^{9}}$ .

$y + C_{1} = e^{- x^{9}} + C_{2}$

Step 2.4

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

$y = e^{- x^{9}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $2$ for $y$ in $y = e^{- x^{9}} + K$ .

$2 = e^{- 0^{9}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $e^{- 0^{9}} + K = 2$ .

$e^{- 0^{9}} + K = 2$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$e^{- 0} + K = 2$

Step 4.2.2

Multiply $- 1$ by $0$ .

$e^{0} + K = 2$

Step 4.2.3

Anything raised to $0$ is $1$ .

$1 + K = 2$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$K = 2 - 1$

Step 4.3.2

Subtract $1$ from $2$ .

$K = 1$

Step 5

Substitute $1$ for $K$ in $y = e^{- x^{9}} + K$ and simplify.

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Step 5.1

Substitute $1$ for $K$ .

$y = e^{- x^{9}} + 1$