Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\sin (y) + y \cos (y)$ .

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

Step 1.2

Cancel the common factor of $\sin (y) + y \cos (y)$ .

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.1.2

Simplify $2 \log (e^{x})$ by moving $2$ inside the logarithm.

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

Step 1.2.1.3

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 1.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(\sin (y) + y \cos (y)) \frac{d y}{d x} = (\sin (y) + y \cos (y)) \frac{x \log (e^{x \cdot 2}) + 1}{\sin (y) + y \cos (y)}$

Step 1.2.1.3.2

Move $2$ to the left of $x$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(\sin (y) + y \cos (y)) d y = (x \log (e^{2 x}) + 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify.

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

Step 2.2.5.2

Simplify.

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Step 2.2.5.2.1

Add $- \cos (y)$ and $\cos (y)$ .

$y \sin (y) + 0 + C_{3} = \int x \log (e^{2 x}) + 1 d x$

Step 2.2.5.2.2

Add $y \sin (y)$ and $0$ .

$y \sin (y) + C_{3} = \int x \log (e^{2 x}) + 1 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $x \log (e^{2 x}) + 1$ as $x (2 x \log (e)) + 1$ .

$y \sin (y) + C_{3} = \int x (2 x \log (e)) + 1 d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y \sin (y) + C_{3} = \int x (2 x \log (e)) d x + \int d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Raise $x$ to the power of $1$ .

$y \sin (y) + C_{3} = \int 2 (x^{1} x) \log (e) d x + \int d x$

Step 2.3.3.2

Raise $x$ to the power of $1$ .

$y \sin (y) + C_{3} = \int 2 (x^{1} x^{1}) \log (e) d x + \int d x$

Step 2.3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y \sin (y) + C_{3} = \int 2 x^{1 + 1} \log (e) d x + \int d x$

Step 2.3.3.4

Add $1$ and $1$ .

$y \sin (y) + C_{3} = \int 2 x^{2} \log (e) d x + \int d x$

Step 2.3.4

Since $2 \log (e)$ is constant with respect to $x$ , move $2 \log (e)$ out of the integral.

$y \sin (y) + C_{3} = 2 \log (e) \int x^{2} d x + \int d x$

Step 2.3.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y \sin (y) + C_{3} = 2 \log (e) (\frac{1}{3} x^{3} + C_{4}) + \int d x$

Step 2.3.6

Apply the constant rule.

$y \sin (y) + C_{3} = 2 \log (e) (\frac{1}{3} x^{3} + C_{4}) + x + C_{5}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y \sin (y) + C_{3} = 2 \log (e) (\frac{x^{3}}{3} + C_{4}) + x + C_{5}$

Step 2.3.7.2

Simplify.

$y \sin (y) + C_{3} = \frac{x^{3} \cdot 2 \log (e)}{3} + x + C_{6}$

Step 2.3.7.3

Move $2$ to the left of $x^{3}$ .

$y \sin (y) + C_{3} = \frac{2 x^{3} \log (e)}{3} + x + C_{6}$

Step 2.3.7.4

Reorder terms.

$y \sin (y) + C_{3} = \frac{2 \log (e)}{3} x^{3} + x + C_{6}$

Step 2.4

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

$y \sin (y) = \frac{2 \log (e)}{3} x^{3} + x + K$