Calculus Examples

$x e^{x^{2}} d x + (y^{5} - 1) d y = 0$

Step 1

Subtract $x e^{x^{2}} d x$ from both sides of the equation.

$(y^{5} - 1) d y = - x e^{x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{5} - 1 d y = \int - x e^{x^{2}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y^{5} d y + \int - 1 d y = \int - x e^{x^{2}} d x$

Step 2.2.2

By the Power Rule, the integral of $y^{5}$ with respect to $y$ is $\frac{1}{6} y^{6}$ .

$\frac{1}{6} y^{6} + C_{1} + \int - 1 d y = \int - x e^{x^{2}} d x$

Step 2.2.3

Apply the constant rule.

$\frac{1}{6} y^{6} + C_{1} - y + C_{2} = \int - x e^{x^{2}} d x$

Step 2.2.4

Simplify.

$\frac{1}{6} y^{6} - y + C_{3} = \int - x e^{x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{6} y^{6} - y + C_{3} = - \int x e^{x^{2}} d x$

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $e^{x^{2}}$ .

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

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 $g (x) = x^{2}$ .

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Step 2.3.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

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 $a$ = $e$ .

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

Step 2.3.2.1.2.3

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

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

Step 2.3.2.1.3

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

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

Step 2.3.2.1.4

Simplify.

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Step 2.3.2.1.4.1

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

$2 e^{x^{2}} x$

Step 2.3.2.1.4.2

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

$2 x e^{x^{2}}$

Step 2.3.2.2

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

$\frac{1}{6} y^{6} - y + C_{3} = - \int \frac{1}{2} d u_{2}$

Step 2.3.3

Apply the constant rule.

$\frac{1}{6} y^{6} - y + C_{3} = - (\frac{1}{2} u_{2} + C_{4})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $- (\frac{1}{2} u_{2} + C_{4})$ as $- \frac{1}{2} u_{2} + C_{4}$ .

$\frac{1}{6} y^{6} - y + C_{3} = - \frac{1}{2} u_{2} + C_{4}$

Step 2.3.4.2

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

$\frac{1}{6} y^{6} - y + C_{3} = - \frac{1}{2} e^{x^{2}} + C_{4}$

Step 2.4

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

$\frac{1}{6} y^{6} - y = - \frac{1}{2} e^{x^{2}} + K$