Calculus Examples

$\int x^{3} \cos (2 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = \cos (2 x)$ .

$x^{3} (\frac{1}{2} \sin (2 x)) - \int \frac{1}{2} \sin (2 x) (3 x^{2}) d x$

Step 2

Simplify.

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Step 2.1

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

$x^{3} \frac{\sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) (3 x^{2}) d x$

Step 2.2

Combine $x^{3}$ and $\frac{\sin (2 x)}{2}$ .

$\frac{x^{3} \sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) (3 x^{2}) d x$

Step 3

Since $\frac{1}{2} \cdot 3$ is constant with respect to $x$ , move $\frac{1}{2} \cdot 3$ out of the integral.

$\frac{x^{3} \sin (2 x)}{2} - (\frac{1}{2} \cdot 3 \int \sin (2 x) (x^{2}) d x)$

Step 4

Combine $\frac{1}{2}$ and $3$ .

$\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} \int \sin (2 x) (x^{2}) d x$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Multiply $2$ by $- 1$ .

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

Cancel the common factors.

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Step 6.7.2.1

Factor $2$ out of $2$ .

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

Step 6.7.2.2

Cancel the common factor.

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

Step 6.7.2.3

Rewrite the expression.

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

Step 6.7.2.4

Divide $- \cos (2 x) x$ by $1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

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

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

Step 12

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 12.1

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

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Step 12.1.1

Differentiate $2 x$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

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

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

Step 15.2

Multiply $2$ by $2$ .

$\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} - \frac{1}{4} \int \sin (u) d u)$

Step 16

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

$\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} - \frac{1}{4} (- \cos (u) + C))$

Step 17

Simplify.

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Step 17.1

Rewrite $\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} - \frac{1}{4} (- \cos (u) + C))$ as $\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4}) + C$ .

$\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4}) + C$

Step 17.2

Simplify.

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Step 17.2.1

To write $- \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{3} \sin (2 x)}{2} - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4}) \cdot \frac{2}{2} + C$

Step 17.2.2

Combine $- \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})$ and $\frac{2}{2}$ .

$\frac{x^{3} \sin (2 x)}{2} + \frac{- \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4}) \cdot 2}{2} + C$

Step 17.2.3

Combine the numerators over the common denominator.

$\frac{x^{3} \sin (2 x) - \frac{3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4}) \cdot 2}{2} + C$

Step 17.2.4

Multiply $2$ by $- 1$ .

$\frac{x^{3} \sin (2 x) - 2 (\frac{3}{2}) (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.5

Combine $- 2$ and $\frac{3}{2}$ .

$\frac{x^{3} \sin (2 x) + \frac{- 2 \cdot 3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.6

Multiply $- 2$ by $3$ .

$\frac{x^{3} \sin (2 x) + \frac{- 6}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.7

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

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Step 17.2.7.1

Factor $2$ out of $- 6$ .

$\frac{x^{3} \sin (2 x) + \frac{2 \cdot - 3}{2} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.7.2

Cancel the common factors.

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Step 17.2.7.2.1

Factor $2$ out of $2$ .

$\frac{x^{3} \sin (2 x) + \frac{2 \cdot - 3}{2 (1)} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.7.2.2

Cancel the common factor.

$\frac{x^{3} \sin (2 x) + \frac{2 \cdot - 3}{2 \cdot 1} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.7.2.3

Rewrite the expression.

$\frac{x^{3} \sin (2 x) + \frac{- 3}{1} (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 17.2.7.2.4

Divide $- 3$ by $1$ .

$\frac{x^{3} \sin (2 x) - 3 (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4})}{2} + C$

Step 18

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

$\frac{x^{3} \sin (2 x) - 3 (- \frac{x^{2} \cos (2 x)}{2} + \frac{x \sin (2 x)}{2} + \frac{\cos (2 x)}{4})}{2} + C$

Step 19

Simplify.

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Step 19.1

Simplify the numerator.

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Step 19.1.1

Apply the distributive property.

$\frac{x^{3} \sin (2 x) - 3 (- \frac{x^{2} \cos (2 x)}{2}) - 3 \frac{x \sin (2 x)}{2} - 3 \frac{\cos (2 x)}{4}}{2} + C$

Step 19.1.2

Simplify.

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Step 19.1.2.1

Multiply $- 3 (- \frac{x^{2} \cos (2 x)}{2})$ .

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Step 19.1.2.1.1

Multiply $- 1$ by $- 3$ .

$\frac{x^{3} \sin (2 x) + 3 \frac{x^{2} \cos (2 x)}{2} - 3 \frac{x \sin (2 x)}{2} - 3 \frac{\cos (2 x)}{4}}{2} + C$

Step 19.1.2.1.2

Combine $3$ and $\frac{x^{2} \cos (2 x)}{2}$ .

$\frac{x^{3} \sin (2 x) + \frac{3 (x^{2} \cos (2 x))}{2} - 3 \frac{x \sin (2 x)}{2} - 3 \frac{\cos (2 x)}{4}}{2} + C$

Step 19.1.2.2

Combine $- 3$ and $\frac{x \sin (2 x)}{2}$ .

$\frac{x^{3} \sin (2 x) + \frac{3 (x^{2} \cos (2 x))}{2} + \frac{- 3 (x \sin (2 x))}{2} - 3 \frac{\cos (2 x)}{4}}{2} + C$

Step 19.1.2.3

Combine $- 3$ and $\frac{\cos (2 x)}{4}$ .

$\frac{x^{3} \sin (2 x) + \frac{3 (x^{2} \cos (2 x))}{2} + \frac{- 3 (x \sin (2 x))}{2} + \frac{- 3 \cos (2 x)}{4}}{2} + C$

Step 19.1.3

Simplify each term.

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Step 19.1.3.1

Move the negative in front of the fraction.

$\frac{x^{3} \sin (2 x) + \frac{3 x^{2} \cos (2 x)}{2} - \frac{3 x \sin (2 x)}{2} + \frac{- 3 \cos (2 x)}{4}}{2} + C$

Step 19.1.3.2

Move the negative in front of the fraction.

$\frac{x^{3} \sin (2 x) + \frac{3 x^{2} \cos (2 x)}{2} - \frac{3 x \sin (2 x)}{2} - \frac{3 \cos (2 x)}{4}}{2} + C$

Step 19.2

Reorder terms.

$\frac{1}{2} (x^{3} \sin (2 x) + \frac{3}{2} x^{2} \cos (2 x) - \frac{3}{2} x \sin (2 x) - \frac{3}{4} \cos (2 x)) + C$