Calculus Examples

$\int x^{2} \sin (5 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sin (5 x)$ .

$x^{2} (- \frac{1}{5} \cos (5 x)) - \int - \frac{1}{5} \cos (5 x) (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\cos (5 x)$ and $\frac{1}{5}$ .

$x^{2} (- \frac{\cos (5 x)}{5}) - \int - \frac{1}{5} \cos (5 x) (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{\cos (5 x)}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} - \int - \frac{1}{5} \cos (5 x) (2 x) d x$

Step 3

Since $- \frac{1}{5} \cdot 2$ is constant with respect to $x$ , move $- \frac{1}{5} \cdot 2$ out of the integral.

$- \frac{x^{2} \cos (5 x)}{5} - (- \frac{1}{5} \cdot 2 \int \cos (5 x) (x) d x)$

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

$- \frac{x^{2} \cos (5 x)}{5} - (- 2 (\frac{1}{5}) \int \cos (5 x) (x) d x)$

Step 4.2

Combine $- 2$ and $\frac{1}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} - (\frac{- 2}{5} \int \cos (5 x) (x) d x)$

Step 4.3

Move the negative in front of the fraction.

$- \frac{x^{2} \cos (5 x)}{5} - (- \frac{2}{5} \int \cos (5 x) (x) d x)$

Step 4.4

Multiply $- 1$ by $- 1$ .

$- \frac{x^{2} \cos (5 x)}{5} + 1 (\frac{2}{5} \int \cos (5 x) (x) d x)$

Step 4.5

Multiply $\frac{2}{5}$ by $1$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} \int \cos (5 x) (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (5 x)$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (x (\frac{1}{5} \sin (5 x)) - \int \frac{1}{5} \sin (5 x) d x)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{5}$ and $\sin (5 x)$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (x \frac{\sin (5 x)}{5} - \int \frac{1}{5} \sin (5 x) d x)$

Step 6.2

Combine $x$ and $\frac{\sin (5 x)}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \int \frac{1}{5} \sin (5 x) d x)$

Step 6.3

Combine $\frac{1}{5}$ and $\sin (5 x)$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \int \frac{\sin (5 x)}{5} d x)$

Step 7

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - (\frac{1}{5} \int \sin (5 x) d x))$

Step 8

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 8.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 8.1.3

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

$5 \cdot 1$

Step 8.1.4

Multiply $5$ by $1$ .

$5$

Step 8.2

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

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{5} \int \sin (u) \frac{1}{5} d u)$

Step 9

Combine $\sin (u)$ and $\frac{1}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{5} \int \frac{\sin (u)}{5} d u)$

Step 10

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{5} (\frac{1}{5} \int \sin (u) d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{5 \cdot 5} \int \sin (u) d u)$

Step 11.2

Multiply $5$ by $5$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{25} \int \sin (u) d u)$

Step 12

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

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{25} (- \cos (u) + C))$

Step 13

Simplify.

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Step 13.1

Rewrite $- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} - \frac{1}{25} (- \cos (u) + C))$ as $- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25}) + C$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25}) + C$

Step 13.2

Simplify.

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Step 13.2.1

To write $\frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25}) \cdot \frac{5}{5} + C$

Step 13.2.2

Combine $\frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})$ and $\frac{5}{5}$ .

$- \frac{x^{2} \cos (5 x)}{5} + \frac{\frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25}) \cdot 5}{5} + C$

Step 13.2.3

Combine the numerators over the common denominator.

$\frac{- x^{2} \cos (5 x) + \frac{2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25}) \cdot 5}{5} + C$

Step 13.2.4

Combine $5$ and $\frac{2}{5}$ .

$\frac{- x^{2} \cos (5 x) + \frac{5 \cdot 2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.5

Multiply $5$ by $2$ .

$\frac{- x^{2} \cos (5 x) + \frac{10}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.6

Cancel the common factor of $10$ and $5$ .

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Step 13.2.6.1

Factor $5$ out of $10$ .

$\frac{- x^{2} \cos (5 x) + \frac{5 \cdot 2}{5} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.6.2

Cancel the common factors.

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Step 13.2.6.2.1

Factor $5$ out of $5$ .

$\frac{- x^{2} \cos (5 x) + \frac{5 \cdot 2}{5 (1)} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.6.2.2

Cancel the common factor.

$\frac{- x^{2} \cos (5 x) + \frac{5 \cdot 2}{5 \cdot 1} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.6.2.3

Rewrite the expression.

$\frac{- x^{2} \cos (5 x) + \frac{2}{1} (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 13.2.6.2.4

Divide $2$ by $1$ .

$\frac{- x^{2} \cos (5 x) + 2 (\frac{x \sin (5 x)}{5} + \frac{\cos (u)}{25})}{5} + C$

Step 14

Replace all occurrences of $u$ with $5 x$ .

$\frac{- x^{2} \cos (5 x) + 2 (\frac{x \sin (5 x)}{5} + \frac{\cos (5 x)}{25})}{5} + C$

Step 15

Simplify.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Apply the distributive property.

$\frac{- x^{2} \cos (5 x) + 2 \frac{x \sin (5 x)}{5} + 2 \frac{\cos (5 x)}{25}}{5} + C$

Step 15.1.2

Combine $2$ and $\frac{x \sin (5 x)}{5}$ .

$\frac{- x^{2} \cos (5 x) + \frac{2 (x \sin (5 x))}{5} + 2 \frac{\cos (5 x)}{25}}{5} + C$

Step 15.1.3

Combine $2$ and $\frac{\cos (5 x)}{25}$ .

$\frac{- x^{2} \cos (5 x) + \frac{2 (x \sin (5 x))}{5} + \frac{2 \cos (5 x)}{25}}{5} + C$

Step 15.1.4

To write $- x^{2} \cos (5 x)$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{\frac{2 x \sin (5 x)}{5} - x^{2} \cos (5 x) \cdot \frac{25}{25} + \frac{2 \cos (5 x)}{25}}{5} + C$

Step 15.1.5

Combine $- x^{2} \cos (5 x)$ and $\frac{25}{25}$ .

$\frac{\frac{2 x \sin (5 x)}{5} + \frac{- x^{2} \cos (5 x) \cdot 25}{25} + \frac{2 \cos (5 x)}{25}}{5} + C$

Step 15.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 x \sin (5 x)}{5} + \frac{- x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.7

To write $\frac{2 x \sin (5 x)}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{2 x \sin (5 x)}{5} \cdot \frac{5}{5} + \frac{- x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.8

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1.8.1

Multiply $\frac{2 x \sin (5 x)}{5}$ by $\frac{5}{5}$ .

$\frac{\frac{2 x \sin (5 x) \cdot 5}{5 \cdot 5} + \frac{- x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.8.2

Multiply $5$ by $5$ .

$\frac{\frac{2 x \sin (5 x) \cdot 5}{25} + \frac{- x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.9

Combine the numerators over the common denominator.

$\frac{\frac{2 x \sin (5 x) \cdot 5 - x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.10

Rewrite $\frac{2 x \sin (5 x) \cdot 5 - x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}$ in a factored form.

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Step 15.1.10.1

Multiply $5$ by $2$ .

$\frac{\frac{10 x \sin (5 x) - x^{2} \cos (5 x) \cdot 25 + 2 \cos (5 x)}{25}}{5} + C$

Step 15.1.10.2

Multiply $25$ by $- 1$ .

$\frac{\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{25}}{5} + C$

Step 15.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{25} \cdot \frac{1}{5} + C$

Step 15.3

Multiply $\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{25} \cdot \frac{1}{5}$ .

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Step 15.3.1

Multiply $\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{25}$ by $\frac{1}{5}$ .

$\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{25 \cdot 5} + C$

Step 15.3.2

Multiply $25$ by $5$ .

$\frac{10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)}{125} + C$

Step 15.4

Reorder terms.

$\frac{1}{125} (10 x \sin (5 x) - 25 x^{2} \cos (5 x) + 2 \cos (5 x)) + C$