Calculus Examples

$(x \cos^{2} (\frac{\frac{π}{4}}{1}) - \frac{π}{4}) d x + 1 d y = 0$

Step 1

Subtract $(x \cos^{2} (\frac{\frac{π}{4}}{1}) - \frac{π}{4}) d x$ from both sides of the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - (x \cos^{2} (\frac{\frac{π}{4}}{1}) - \frac{π}{4}) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - (x \cos^{2} (\frac{\frac{π}{4}}{1}) - \frac{π}{4}) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Divide $\frac{π}{4}$ by $1$ .

$y + C_{1} = \int - (x \cos^{2} (\frac{π}{4}) - \frac{π}{4}) d x$

Step 2.3.2

Multiply $- (x \cos^{2} (\frac{π}{4}) - \frac{π}{4})$ .

$y + C_{1} = \int - (x \cos^{2} (\frac{π}{4})) - - \frac{π}{4} d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$y + C_{1} = \int - x \cos^{2} (\frac{π}{4}) + 1 \frac{π}{4} d x$

Step 2.3.3.2

Multiply $\frac{π}{4}$ by $1$ .

$y + C_{1} = \int - x \cos^{2} (\frac{π}{4}) + \frac{π}{4} d x$

Step 2.3.4

Split the single integral into multiple integrals.

$y + C_{1} = \int - x \cos^{2} (\frac{π}{4}) d x + \int \frac{π}{4} d x$

Step 2.3.5

Since $- \cos^{2} (\frac{π}{4})$ is constant with respect to $x$ , move $- \cos^{2} (\frac{π}{4})$ out of the integral.

$y + C_{1} = - \cos^{2} (\frac{π}{4}) \int x d x + \int \frac{π}{4} d x$

Step 2.3.6

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

$y + C_{1} = - \cos^{2} (\frac{π}{4}) (\frac{1}{2} x^{2} + C_{2}) + \int \frac{π}{4} d x$

Step 2.3.7

Apply the constant rule.

$y + C_{1} = - \cos^{2} (\frac{π}{4}) (\frac{1}{2} x^{2} + C_{2}) + \frac{π}{4} x + C_{3}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

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

$y + C_{1} = - \cos^{2} (\frac{π}{4}) (\frac{x^{2}}{2} + C_{2}) + \frac{π}{4} x + C_{3}$

Step 2.3.8.2

Simplify.

$y + C_{1} = - {(\frac{\sqrt{2}}{2})}^{2} \frac{x^{2}}{2} + \frac{π}{4} x + C_{4}$

Step 2.3.9

Reorder terms.

$y + C_{1} = - {(\frac{\sqrt{2}}{2})}^{2} \frac{1}{2} x^{2} + \frac{π}{4} x + C_{4}$

Step 2.3.10

Reorder terms.

$y + C_{1} = - \frac{1}{2} {(\frac{\sqrt{2}}{2})}^{2} x^{2} + \frac{π}{4} x + C_{4}$

Step 2.4

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

$y = - \frac{1}{2} {(\frac{\sqrt{2}}{2})}^{2} x^{2} + \frac{π}{4} x + K$