Calculus Examples

$\frac{d y}{d x} = \frac{4}{1 + x^{2}}$ , $y (1) = \frac{π}{2}$

Step 1

Rewrite the equation.

$d y = \frac{4}{1 + x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{4}{1 + x^{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{4}{1 + x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int \frac{1}{1 + x^{2}} d x$

Step 2.3.2

Rewrite $1$ as $1^{2}$ .

$y + C_{1} = 4 \int \frac{1}{1^{2} + x^{2}} d x$

Step 2.3.3

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{2}$ .

$y + C_{1} = 4 (\arctan (x) + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = 4 \arctan (x) + C_{2}$

Step 2.4

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

$y = 4 \arctan (x) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $\frac{π}{2}$ for $y$ in $y = 4 \arctan (x) + K$ .

$\frac{π}{2} = 4 \arctan (1) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $4 \arctan (1) + K = \frac{π}{2}$ .

$4 \arctan (1) + K = \frac{π}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$4 \frac{π}{4} + K = \frac{π}{2}$

Step 4.2.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.2.1

Cancel the common factor.

$4 \frac{π}{4} + K = \frac{π}{2}$

Step 4.2.1.2.2

Rewrite the expression.

$π + K = \frac{π}{2}$

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 $π$ from both sides of the equation.

$K = \frac{π}{2} - π$

Step 4.3.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$K = \frac{π}{2} - π \cdot \frac{2}{2}$

Step 4.3.3

Combine $- π$ and $\frac{2}{2}$ .

$K = \frac{π}{2} + \frac{- π \cdot 2}{2}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{π - π \cdot 2}{2}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $2$ by $- 1$ .

$K = \frac{π - 2 π}{2}$

Step 4.3.5.2

Subtract $2 π$ from $π$ .

$K = \frac{- π}{2}$

Step 4.3.6

Move the negative in front of the fraction.

$K = - \frac{π}{2}$

Step 5

Substitute $- \frac{π}{2}$ for $K$ in $y = 4 \arctan (x) + K$ and simplify.

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Step 5.1

Substitute $- \frac{π}{2}$ for $K$ .

$y = 4 \arctan (x) - \frac{π}{2}$