Calculus Examples

$\frac{d y}{d x} = e^{2 x} \cos^{2} (y)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\cos^{2} (y)}$ .

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (e^{2 x} \cos^{2} (y))$

Step 1.2

Cancel the common factor of $\cos^{2} (y)$ .

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Step 1.2.1

Factor $\cos^{2} (y)$ out of $e^{2 x} \cos^{2} (y)$ .

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (\cos^{2} (y) e^{2 x})$

Step 1.2.2

Cancel the common factor.

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (\cos^{2} (y) e^{2 x})$

Step 1.2.3

Rewrite the expression.

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = e^{2 x}$

Step 1.3

Rewrite the equation.

$\frac{1}{\cos^{2} (y)} d y = e^{2 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\cos^{2} (y)} d y = \int e^{2 x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Convert from $\frac{1}{\cos^{2} (y)}$ to $\sec^{2} (y)$ .

$\int \sec^{2} (y) d y = \int e^{2 x} d x$

Step 2.2.2

Since the derivative of $\tan (y)$ is $\sec^{2} (y)$ , the integral of $\sec^{2} (y)$ is $\tan (y)$ .

$\tan (y) + C_{1} = \int e^{2 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $2 x$ .

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

Step 2.3.1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.3.1.1.3

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

$2 \cdot 1$

Step 2.3.1.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

$\tan (y) + C_{1} = \int e^{u} \frac{1}{2} d u$

Step 2.3.2

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

$\tan (y) + C_{1} = \int \frac{e^{u}}{2} d u$

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\tan (y) + C_{1} = \frac{1}{2} \int e^{u} d u$

Step 2.3.4

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

$\tan (y) + C_{1} = \frac{1}{2} (e^{u} + C_{2})$

Step 2.3.5

Simplify.

$\tan (y) + C_{1} = \frac{1}{2} e^{u} + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $2 x$ .

$\tan (y) + C_{1} = \frac{1}{2} e^{2 x} + C_{2}$

Step 2.4

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

$\tan (y) = \frac{1}{2} e^{2 x} + K$

Step 3

Solve for $y$ .

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Step 3.1

Simplify $\frac{1}{2} \cdot e^{2 x} + K$ .

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Step 3.1.1

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

$\tan (y) = \frac{e^{2 x}}{2} + K$

Step 3.1.2

Reorder $\frac{e^{2 x}}{2}$ and $K$ .

$\tan (y) = K + \frac{e^{2 x}}{2}$

Step 3.2

Take the inverse tangent of both sides of the equation to extract $y$ from inside the tangent.

$y = \arctan (K + \frac{e^{2 x}}{2})$