Calculus Examples

$\frac{d y}{d x} = 2 x \sec (y)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sec (y)}$ .

$\frac{1}{\sec (y)} \frac{d y}{d x} = \frac{1}{\sec (y)} (2 x \sec (y))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sec (y)} \frac{d y}{d x} = 2 \frac{1}{\sec (y)} x \sec (y)$

Step 1.2.2

Rewrite $\sec (y)$ in terms of sines and cosines.

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

Step 1.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

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

Step 1.2.4

Multiply $\cos (y)$ by $1$ .

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

Step 1.2.5

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.2.5.1

Add parentheses.

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

Step 1.2.5.2

Add parentheses.

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

Step 1.2.5.3

Reorder $\cos (y)$ and $x \sec (y)$ .

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

Step 1.2.5.4

Add parentheses.

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

Step 1.2.5.5

Rewrite $2 \cos (y) x \sec (y)$ in terms of sines and cosines.

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

Step 1.2.5.6

Cancel the common factors.

$\frac{1}{\sec (y)} \frac{d y}{d x} = 2 (x \cdot 1)$

Step 1.2.6

Multiply $x$ by $1$ .

$\frac{1}{\sec (y)} \frac{d y}{d x} = 2 x$

Step 1.3

Rewrite the equation.

$\frac{1}{\sec (y)} d y = 2 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sec (y)} d y = \int 2 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Rewrite $\sec (y)$ in terms of sines and cosines.

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

Step 2.2.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

$\int 1 \cos (y) d y = \int 2 x d x$

Step 2.2.1.3

Multiply $\cos (y)$ by $1$ .

$\int \cos (y) d y = \int 2 x d x$

Step 2.2.2

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

$\sin (y) + C_{1} = \int 2 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\sin (y) + C_{1} = 2 \int x d x$

Step 2.3.2

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C_{2})$ as $2 (\frac{1}{2}) x^{2} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

$\sin (y) + C_{1} = 1 x^{2} + C_{2}$

Step 2.3.3.2.3

Multiply $x^{2}$ by $1$ .

$\sin (y) + C_{1} = x^{2} + C_{2}$

Step 2.4

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

$\sin (y) = x^{2} + K$

Step 3

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

$y = \arcsin (x^{2} + K)$