Calculus Examples

$\int x \sin (2 x) d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$\int \frac{u}{2} \sin (u) \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{u \sin (u)}{2} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u \sin (u)}{2}$ by $\frac{1}{2}$ .

$\int \frac{u \sin (u)}{2 \cdot 2} d u$

Step 2.3

Multiply $2$ by $2$ .

$\int \frac{u \sin (u)}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int u \sin (u) d u$

Step 4

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

$\frac{1}{4} (u (- \cos (u)) - \int - \cos (u) d u)$

Step 5

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

$\frac{1}{4} (u (- \cos (u)) - - \int \cos (u) d u)$

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} (u (- \cos (u)) + 1 \int \cos (u) d u)$

Step 6.2

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

$\frac{1}{4} (u (- \cos (u)) + \int \cos (u) d u)$

Step 7

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

$\frac{1}{4} (u (- \cos (u)) + \sin (u) + C)$

Step 8

Rewrite $\frac{1}{4} (u (- \cos (u)) + \sin (u) + C)$ as $\frac{1}{4} (- u \cos (u) + \sin (u)) + C$ .

$\frac{1}{4} (- u \cos (u) + \sin (u)) + C$

Step 9

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

$\frac{1}{4} (- (2 x) \cos (2 x) + \sin (2 x)) + C$

Step 10

Simplify.

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Step 10.1

Multiply $2$ by $- 1$ .

$\frac{1}{4} (- 2 x \cos (2 x) + \sin (2 x)) + C$

Step 10.2

Apply the distributive property.

$\frac{1}{4} (- 2 x \cos (2 x)) + \frac{1}{4} \sin (2 x) + C$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (- 2 x \cos (2 x)) + \frac{1}{4} \sin (2 x) + C$

Step 10.3.2

Factor $2$ out of $- 2 x \cos (2 x)$ .

$\frac{1}{2 (2)} (2 (- x \cos (2 x))) + \frac{1}{4} \sin (2 x) + C$

Step 10.3.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 (- x \cos (2 x))) + \frac{1}{4} \sin (2 x) + C$

Step 10.3.4

Rewrite the expression.

$\frac{1}{2} (- x \cos (2 x)) + \frac{1}{4} \sin (2 x) + C$

Step 10.4

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

$- \frac{x}{2} \cos (2 x) + \frac{1}{4} \sin (2 x) + C$

Step 10.5

Combine $\cos (2 x)$ and $\frac{x}{2}$ .

$- \frac{\cos (2 x) x}{2} + \frac{1}{4} \sin (2 x) + C$

Step 10.6

Combine $\frac{1}{4}$ and $\sin (2 x)$ .

$- \frac{\cos (2 x) x}{2} + \frac{\sin (2 x)}{4} + C$

Step 10.7

Reorder factors in $- \frac{\cos (2 x) x}{2} + \frac{\sin (2 x)}{4}$ .

$- \frac{x \cos (2 x)}{2} + \frac{\sin (2 x)}{4} + C$

Step 11

Reorder terms.

$- \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (2 x) + C$