Calculus Examples

$\frac{d y}{d x} = \frac{6 x^{5} - 2 x + 1}{\cos (y) + e^{y}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\cos (y) + e^{y}$ .

$(\cos (y) + e^{y}) \frac{d y}{d x} = (\cos (y) + e^{y}) \frac{6 x^{5} - 2 x + 1}{\cos (y) + e^{y}}$

Step 1.2

Cancel the common factor of $\cos (y) + e^{y}$ .

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Step 1.2.1

Cancel the common factor.

$(\cos (y) + e^{y}) \frac{d y}{d x} = (\cos (y) + e^{y}) \frac{6 x^{5} - 2 x + 1}{\cos (y) + e^{y}}$

Step 1.2.2

Rewrite the expression.

$(\cos (y) + e^{y}) \frac{d y}{d x} = 6 x^{5} - 2 x + 1$

Step 1.3

Rewrite the equation.

$(\cos (y) + e^{y}) d y = (6 x^{5} - 2 x + 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (y) + e^{y} d y = \int 6 x^{5} - 2 x + 1 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int \cos (y) d y + \int e^{y} d y = \int 6 x^{5} - 2 x + 1 d x$

Step 2.2.2

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

$\sin (y) + C_{1} + \int e^{y} d y = \int 6 x^{5} - 2 x + 1 d x$

Step 2.2.3

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$\sin (y) + C_{1} + e^{y} + C_{2} = \int 6 x^{5} - 2 x + 1 d x$

Step 2.2.4

Simplify.

$\sin (y) + e^{y} + C_{3} = \int 6 x^{5} - 2 x + 1 d x$

Step 2.2.5

Reorder terms.

$e^{y} + \sin (y) + C_{3} = \int 6 x^{5} - 2 x + 1 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$e^{y} + \sin (y) + C_{3} = \int 6 x^{5} d x + \int - 2 x d x + \int d x$

Step 2.3.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{y} + \sin (y) + C_{3} = 6 \int x^{5} d x + \int - 2 x d x + \int d x$

Step 2.3.3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$e^{y} + \sin (y) + C_{3} = 6 (\frac{1}{6} x^{6} + C_{4}) + \int - 2 x d x + \int d x$

Step 2.3.4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$e^{y} + \sin (y) + C_{3} = 6 (\frac{1}{6} x^{6} + C_{4}) - 2 \int x d x + \int d x$

Step 2.3.5

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

$e^{y} + \sin (y) + C_{3} = 6 (\frac{1}{6} x^{6} + C_{4}) - 2 (\frac{1}{2} x^{2} + C_{5}) + \int d x$

Step 2.3.6

Apply the constant rule.

$e^{y} + \sin (y) + C_{3} = 6 (\frac{1}{6} x^{6} + C_{4}) - 2 (\frac{1}{2} x^{2} + C_{5}) + x + C_{6}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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Step 2.3.7.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$e^{y} + \sin (y) + C_{3} = 6 (\frac{x^{6}}{6} + C_{4}) - 2 (\frac{1}{2} x^{2} + C_{5}) + x + C_{6}$

Step 2.3.7.1.2

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

$e^{y} + \sin (y) + C_{3} = 6 (\frac{x^{6}}{6} + C_{4}) - 2 (\frac{x^{2}}{2} + C_{5}) + x + C_{6}$

Step 2.3.7.2

Simplify.

$e^{y} + \sin (y) + C_{3} = x^{6} - x^{2} + x + C_{7}$

Step 2.4

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

$e^{y} + \sin (y) = x^{6} - x^{2} + x + K$