Calculus Examples

$\frac{d y}{d x} = x - \sin (x)$ , $y (π) = 3$

Step 1

Rewrite the equation.

$d y = (x - \sin (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x - \sin (x) d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5.2.2

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

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $π$ for $x$ and $3$ for $y$ in $y = \frac{1}{2} x^{2} + \cos (x) + K$ .

$3 = \frac{1}{2} π^{2} + \cos (π) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{2} \cdot π^{2} + \cos (π) + K = 3$ .

$\frac{1}{2} \cdot π^{2} + \cos (π) + K = 3$

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

Combine $\frac{1}{2}$ and $π^{2}$ .

$\frac{π^{2}}{2} + \cos (π) + K = 3$

Step 4.2.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{π^{2}}{2} - \cos (0) + K = 3$

Step 4.2.1.3

The exact value of $\cos (0)$ is $1$ .

$\frac{π^{2}}{2} - 1 \cdot 1 + K = 3$

Step 4.2.1.4

Multiply $- 1$ by $1$ .

$\frac{π^{2}}{2} - 1 + K = 3$

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 $\frac{π^{2}}{2}$ from both sides of the equation.

$- 1 + K = 3 - \frac{π^{2}}{2}$

Step 4.3.2

Add $1$ to both sides of the equation.

$K = 3 - \frac{π^{2}}{2} + 1$

Step 4.3.3

Add $3$ and $1$ .

$K = - \frac{π^{2}}{2} + 4$

Step 5

Substitute $- \frac{π^{2}}{2} + 4$ for $K$ in $y = \frac{1}{2} x^{2} + \cos (x) + K$ and simplify.

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Step 5.1

Substitute $- \frac{π^{2}}{2} + 4$ for $K$ .

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

Step 5.2

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

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