Calculus Examples

$(6 y - \sec^{2} (y)) d y - (1 + \sin (x)) d x = 0$

Step 1

Add $(1 + \sin (x)) d x$ to both sides of the equation.

$(6 y - \sec^{2} (y)) d y = (1 + \sin (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Since the derivative of $\tan (y)$ is $\sec^{2} (y)$ , the integral of $\sec^{2} (y)$ is $\tan (y)$ .

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Simplify.

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

Step 2.2.6.2

Simplify.

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Step 2.2.6.2.1

Combine $6$ and $\frac{1}{2}$ .

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

Step 2.2.6.2.2

Cancel the common factor of $6$ and $2$ .

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Step 2.2.6.2.2.1

Factor $2$ out of $6$ .

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

Step 2.2.6.2.2.2

Cancel the common factors.

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Step 2.2.6.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.6.2.2.2.2

Cancel the common factor.

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

Step 2.2.6.2.2.2.3

Rewrite the expression.

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

Step 2.2.6.2.2.2.4

Divide $3$ by $1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

$3 y^{2} - \tan (y) + C_{3} = x + C_{4} + \int \sin (x) d x$

Step 2.3.3

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

$3 y^{2} - \tan (y) + C_{3} = x + C_{4} - \cos (x) + C_{5}$

Step 2.3.4

Simplify.

$3 y^{2} - \tan (y) + C_{3} = x - \cos (x) + C_{6}$

Step 2.4

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

$3 y^{2} - \tan (y) = x - \cos (x) + K$