Calculus Examples

$\frac{d y}{d x} = e^{x} \sin (x) + x \cos (x)$

Step 1

Rewrite the equation.

$d y = (e^{x} \sin (x) + x \cos (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int e^{x} \sin (x) + x \cos (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int e^{x} \sin (x) + x \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int e^{x} \sin (x) d x + \int x \cos (x) d x$

Step 2.3.2

Reorder $e^{x}$ and $\sin (x)$ .

$y + C_{1} = \int \sin (x) e^{x} d x + \int x \cos (x) d x$

Step 2.3.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (x)$ and $d v = e^{x}$ .

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

Step 2.3.4

Reorder $e^{x}$ and $\cos (x)$ .

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

Step 2.3.5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (x)$ and $d v = e^{x}$ .

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

Step 2.3.6

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

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

Step 2.3.7

Simplify by multiplying through.

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Step 2.3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.7.2

Multiply $\int e^{x} (\sin (x)) d x$ by $1$ .

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

Step 2.3.7.3

Apply the distributive property.

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

Step 2.3.8

Solving for $\int e^{x} \sin (x) d x$ , we find that $\int e^{x} \sin (x) d x$ = $\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2}$ .

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

Step 2.3.9

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (x)$ .

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

Step 2.3.10

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

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

Step 2.3.11

Simplify.

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Step 2.3.11.1

Simplify.

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Step 2.3.11.1.1

Add $\frac{\sin (x) e^{x} - \cos (x) e^{x}}{2}$ and $x \sin (x)$ .

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Step 2.3.11.1.1.1

Move $\cos (x)$ .

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

Step 2.3.11.1.1.2

To write $x \sin (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.11.1.1.3

Combine $x \sin (x)$ and $\frac{2}{2}$ .

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

Step 2.3.11.1.1.4

Combine the numerators over the common denominator.

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

Step 2.3.11.1.2

Move $2$ to the left of $x \sin (x)$ .

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

Step 2.3.11.2

Simplify.

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

Step 2.3.11.3

Simplify.

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Step 2.3.11.3.1

To write $\cos (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.11.3.2

Combine $\cos (x)$ and $\frac{2}{2}$ .

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

Step 2.3.11.3.3

Combine the numerators over the common denominator.

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

Step 2.3.11.3.4

Move $2$ to the left of $\cos (x)$ .

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

Step 2.3.11.4

Reorder terms.

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

Step 2.4

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

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