Calculus Examples

$x \sin^{2} (x)$

Step 1

Write $x \sin^{2} (x)$ as a function.

$f (x) = x \sin^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

$x (\frac{1}{2} x - \frac{1}{4} \sin (2 x)) - \int \frac{1}{2} x - \frac{1}{4} \sin (2 x) d x$

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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 9.1

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

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Step 9.1.1

Differentiate $2 x$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

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

Step 13.1.2

Multiply $2$ by $4$ .

$x (\frac{x}{2} - \frac{\sin (2 x)}{4}) - (\frac{1}{2} (\frac{1}{2} x^{2} + C) - \frac{1}{8} (- \cos (u) + C))$

Step 13.1.3

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

$x (\frac{x}{2} - \frac{\sin (2 x)}{4}) - (\frac{1}{2} (\frac{x^{2}}{2} + C) - \frac{1}{8} (- \cos (u) + C))$

Step 13.2

Simplify.

$\frac{x^{2}}{2} - x \frac{\sin (2 x)}{4} - (\frac{1}{2} \cdot \frac{x^{2}}{2} - \frac{1}{8} (- \cos (u))) + C$

Step 13.3

Simplify.

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Step 13.3.1

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

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{1}{2} \cdot \frac{x^{2}}{2} - \frac{1}{8} (- \cos (u))) + C$

Step 13.3.2

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

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{2 \cdot 2} - \frac{1}{8} (- \cos (u))) + C$

Step 13.3.3

Multiply $2$ by $2$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{4} - \frac{1}{8} (- \cos (u))) + C$

Step 13.3.4

Multiply $- 1$ by $- 1$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{4} + 1 (\frac{1}{8}) \cos (u)) + C$

Step 13.3.5

Multiply $\frac{1}{8}$ by $1$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{4} + \frac{1}{8} \cos (u)) + C$

Step 13.3.6

Combine $\frac{1}{8}$ and $\cos (u)$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{4} + \frac{\cos (u)}{8}) + C$

Step 13.3.7

To write $- (\frac{x^{2}}{4} + \frac{\cos (u)}{8})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} - (\frac{x^{2}}{4} + \frac{\cos (u)}{8}) \cdot \frac{4}{4} + C$

Step 13.3.8

Combine $- (\frac{x^{2}}{4} + \frac{\cos (u)}{8})$ and $\frac{4}{4}$ .

$\frac{x^{2}}{2} - \frac{\sin (2 x) x}{4} + \frac{- (\frac{x^{2}}{4} + \frac{\cos (u)}{8}) \cdot 4}{4} + C$

Step 13.3.9

Combine the numerators over the common denominator.

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - (\frac{x^{2}}{4} + \frac{\cos (u)}{8}) \cdot 4}{4} + C$

Step 13.3.10

Multiply $4$ by $- 1$ .

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - 4 (\frac{x^{2}}{4} + \frac{\cos (u)}{8})}{4} + C$

Step 14

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

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - 4 (\frac{x^{2}}{4} + \frac{\cos (2 x)}{8})}{4} + C$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - 4 \frac{x^{2}}{4} - 4 \frac{\cos (2 x)}{8}}{4} + C$

Step 15.2

Cancel the common factor of $4$ .

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Step 15.2.1

Factor $4$ out of $- 4$ .

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x + 4 (- 1) \frac{x^{2}}{4} - 4 \frac{\cos (2 x)}{8}}{4} + C$

Step 15.2.2

Cancel the common factor.

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x + 4 \cdot - 1 \frac{x^{2}}{4} - 4 \frac{\cos (2 x)}{8}}{4} + C$

Step 15.2.3

Rewrite the expression.

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - x^{2} - 4 \frac{\cos (2 x)}{8}}{4} + C$

Step 15.3

Cancel the common factor of $4$ .

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Step 15.3.1

Factor $4$ out of $- 4$ .

$\frac{x^{2}}{2} + \frac{- \sin (2 x) x - x^{2} + 4 (- 1) \frac{\cos (2 x)}{8}}{4} + C$

Step 15.3.2

Factor $4$ out of $8$ .

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

Step 15.3.3

Cancel the common factor.

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

Step 15.3.4

Rewrite the expression.

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

Step 16

Simplify.

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Step 16.1

Factor $- 1$ out of $- \sin (2 x) x$ .

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

Step 16.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 16.3

Factor $- 1$ out of $- (\sin (2 x) x) - (x^{2})$ .

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

Step 16.4

Factor $- 1$ out of $- \frac{\cos (2 x)}{2}$ .

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

Step 16.5

Factor $- 1$ out of $- (\sin (2 x) x + x^{2}) - (\frac{\cos (2 x)}{2})$ .

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

Step 16.6

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

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

Step 16.7

Move the negative in front of the fraction.

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

Step 16.8

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

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

Step 16.9

Reorder terms.

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

Step 17

The answer is the antiderivative of the function $f (x) = x \sin^{2} (x)$ .

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