Calculus Examples

$\frac{d y}{d x} = \sec^{2} (x)$ , $y (0) = 3$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

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

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

Step 2.4

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

$y = \tan (x) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $3$ for $y$ in $y = \tan (x) + K$ .

$3 = \tan (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\tan (0) + K = 3$ .

$\tan (0) + K = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\tan (0) + K$ .

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Step 4.2.1.1

The exact value of $\tan (0)$ is $0$ .

$0 + K = 3$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 3$

Step 5

Substitute $3$ for $K$ in $y = \tan (x) + K$ and simplify.

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Step 5.1

Substitute $3$ for $K$ .

$y = \tan (x) + 3$