Calculus Examples

$\frac{d y}{d x} = x \sin (x)$

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

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.2

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

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

Step 2.3.4

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

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

Step 2.3.5

Rewrite $x (- \cos (x)) + \sin (x) + C_{2}$ as $- x \cos (x) + \sin (x) + C_{2}$ .

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

Step 2.4

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

$y = - x \cos (x) + \sin (x) + K$