Calculus Examples

$\int x^{2} \cos (2 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \cos (2 x)$ .

$x^{2} (\frac{1}{2} \sin (2 x)) - \int \frac{1}{2} \sin (2 x) (2 x) d x$

Step 2

Simplify.

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Step 2.1

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

$x^{2} \frac{\sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) (2 x) d x$

Step 2.2

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

$\frac{x^{2} \sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) (2 x) d x$

Step 3

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

$\frac{x^{2} \sin (2 x)}{2} - (\frac{1}{2} \cdot 2 \int \sin (2 x) (x) d x)$

Step 4

Simplify.

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Step 4.1

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

$\frac{x^{2} \sin (2 x)}{2} - (\frac{2}{2} \int \sin (2 x) (x) d x)$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

$\frac{x^{2} \sin (2 x)}{2} - (\frac{2}{2} \int \sin (2 x) (x) d x)$

Step 4.2.2

Rewrite the expression.

$\frac{x^{2} \sin (2 x)}{2} - (1 \int \sin (2 x) (x) d x)$

Step 4.3

Multiply $\int \sin (2 x) (x) d x$ by $1$ .

$\frac{x^{2} \sin (2 x)}{2} - \int \sin (2 x) (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (2 x)$ .

$\frac{x^{2} \sin (2 x)}{2} - (x (- \frac{1}{2} \cos (2 x)) - \int - \frac{1}{2} \cos (2 x) d x)$

Step 6

Simplify.

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Step 6.1

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

$\frac{x^{2} \sin (2 x)}{2} - (x (- \frac{\cos (2 x)}{2}) - \int - \frac{1}{2} \cos (2 x) d x)$

Step 6.2

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

$\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} - \int - \frac{1}{2} \cos (2 x) d x)$

Step 6.3

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

$\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} - \int - \frac{\cos (2 x)}{2} d x)$

Step 7

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

$\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} - - \int \frac{\cos (2 x)}{2} d x)$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int \frac{\cos (2 x)}{2} d x$ by $1$ .

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

Step 9

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

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

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $2 x$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} + \frac{1}{2 \cdot 2} \int \cos (u) d u)$

Step 13.2

Multiply $2$ by $2$ .

$\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} + \frac{1}{4} \int \cos (u) d u)$

Step 14

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

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

Step 15

Simplify.

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Step 15.1

Rewrite $\frac{x^{2} \sin (2 x)}{2} - (- \frac{x \cos (2 x)}{2} + \frac{1}{4} (\sin (u) + C))$ as $\frac{1}{2} x^{2} \sin (2 x) - (- \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (u)) + C$ .

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

Step 15.2

Simplify.

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Step 15.2.1

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

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

Step 15.2.2

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

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

Step 15.2.3

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

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

Step 15.2.4

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

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

Step 15.2.5

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

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

Step 15.2.6

To write $- (- \frac{\cos (2 x) x}{2} + \frac{\sin (u)}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.2.7

Combine $- (- \frac{\cos (2 x) x}{2} + \frac{\sin (u)}{4})$ and $\frac{2}{2}$ .

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

Step 15.2.8

Combine the numerators over the common denominator.

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

Step 15.2.9

Multiply $2$ by $- 1$ .

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Apply the distributive property.

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

Step 17.2

Cancel the common factor of $2$ .

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Step 17.2.1

Move the leading negative in $- \frac{\cos (2 x) x}{2}$ into the numerator.

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

Step 17.2.2

Factor $2$ out of $- 2$ .

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

Step 17.2.3

Cancel the common factor.

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

Step 17.2.4

Rewrite the expression.

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

Step 17.3

Multiply $- 1$ by $- 1$ .

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

Step 17.4

Multiply $\cos (2 x)$ by $1$ .

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

Step 17.5

Cancel the common factor of $2$ .

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Step 17.5.1

Factor $2$ out of $- 2$ .

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

Step 17.5.2

Factor $2$ out of $4$ .

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

Step 17.5.3

Cancel the common factor.

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

Step 17.5.4

Rewrite the expression.

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

Step 17.6

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

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

Step 18

Simplify.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

To write $x^{2} \sin (2 x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 18.1.2

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

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

Step 18.1.3

Combine the numerators over the common denominator.

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

Step 18.1.4

To write $x \cos (2 x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 18.1.5

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

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

Step 18.1.6

Combine the numerators over the common denominator.

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

Step 18.1.7

Rewrite $\frac{x \cos (2 x) \cdot 2 + x^{2} \sin (2 x) \cdot 2 - \sin (2 x)}{2}$ in a factored form.

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Step 18.1.7.1

Move $2$ to the left of $x \cos (2 x)$ .

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

Step 18.1.7.2

Move $2$ to the left of $x^{2} \sin (2 x)$ .

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

Step 18.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 18.3

Multiply $\frac{2 x \cos (2 x) + 2 x^{2} \sin (2 x) - \sin (2 x)}{2} \cdot \frac{1}{2}$ .

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Step 18.3.1

Multiply $\frac{2 x \cos (2 x) + 2 x^{2} \sin (2 x) - \sin (2 x)}{2}$ by $\frac{1}{2}$ .

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

Step 18.3.2

Multiply $2$ by $2$ .

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

Step 18.4

Reorder terms.

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