Calculus Examples

$\int \cos (3 y) - 2 x d x \int \sin (2 x) + 2 yd y d x$

Step 1

Evaluate $\int \cos (3 y) - 2 x d x$ .

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Step 1.1

Split the single integral into multiple integrals.

$(\int \cos (3 y) d x + \int - 2 x d x) \int \sin (2 x) + 2 yd y d x$

Step 1.2

Apply the constant rule.

$(\cos (3 y) x + C + \int - 2 x d x) \int \sin (2 x) + 2 yd y d x$

Step 1.3

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

$(\cos (3 y) x + C - 2 \int x d x) \int \sin (2 x) + 2 yd y d x$

Step 1.4

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

$(\cos (3 y) x + C - 2 (\frac{1}{2} x^{2} + C)) \int \sin (2 x) + 2 yd y d x$

Step 1.5

Simplify.

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Step 1.5.1

Simplify.

$(\cos (3 y) x - 2 (\frac{1}{2}) x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2

Simplify.

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Step 1.5.2.1

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

$(\cos (3 y) x + \frac{- 2}{2} x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2.2

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

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Step 1.5.2.2.1

Factor $2$ out of $- 2$ .

$(\cos (3 y) x + \frac{2 \cdot - 1}{2} x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2.2.2

Cancel the common factors.

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Step 1.5.2.2.2.1

Factor $2$ out of $2$ .

$(\cos (3 y) x + \frac{2 \cdot - 1}{2 (1)} x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2.2.2.2

Cancel the common factor.

$(\cos (3 y) x + \frac{2 \cdot - 1}{2 \cdot 1} x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2.2.2.3

Rewrite the expression.

$(\cos (3 y) x + \frac{- 1}{1} x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 1.5.2.2.2.4

Divide $- 1$ by $1$ .

$(\cos (3 y) x - x^{2} + C) \int \sin (2 x) + 2 yd y d x$

Step 2

Evaluate $\int \sin (2 x) + 2 yd y d x$ .

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Step 2.1

Split the single integral into multiple integrals.

$(\cos (3 y) x - x^{2} + C) (\int \sin (2 x) d x + \int 2 yd y d x)$

Step 2.2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 2.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.2

Rewrite the problem using $u$ and $d u$ .

$(\cos (3 y) x - x^{2} + C) (\int \sin (u) \frac{1}{2} d u + \int 2 yd y d x)$

Step 2.3

Combine $\sin (u)$ and $\frac{1}{2}$ .

$(\cos (3 y) x - x^{2} + C) (\int \frac{\sin (u)}{2} d u + \int 2 yd y d x)$

Step 2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$(\cos (3 y) x - x^{2} + C) (\frac{1}{2} \int \sin (u) d u + \int 2 yd y d x)$

Step 2.5

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

$(\cos (3 y) x - x^{2} + C) (\frac{1}{2} (- \cos (u) + C) + \int 2 yd y d x)$

Step 2.6

Apply the constant rule.

$(\cos (3 y) x - x^{2} + C) (\frac{1}{2} (- \cos (u) + C) + 2 yd y x + C)$

Step 2.7

Simplify.

$(\cos (3 y) x - x^{2} + C) (- \frac{\cos (u)}{2} + 2 yd y x + C)$

Step 2.8

Replace all occurrences of $u$ with $2 x$ .

$(\cos (3 y) x - x^{2} + C) (- \frac{\cos (2 x)}{2} + 2 yd y x + C)$

Step 2.9

Reorder terms.

$(\cos (3 y) x - x^{2} + C) (- \frac{1}{2} \cos (2 x) + 2 yd y x + C)$

Step 3

Simplify.

$(\cos (3 y) x - x^{2}) (- \frac{1}{2} \cos (2 x) + 2 yd y x) + C$