Calculus Examples

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

Step 1

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

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

Step 2

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 3

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 4

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 5

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 6

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 6.1

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

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Step 6.1.1

Differentiate $2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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 7

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 8

Simplify.

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Step 8.1

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 8.2

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

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

Step 9

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 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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 10.1.2

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 10.1.3

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 10.2

Simplify.

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

Step 10.3

Simplify.

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Step 10.3.1

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

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

Step 10.3.2

Multiply $2$ by $2$ .

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

Step 10.3.3

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

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

Step 10.3.4

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

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

Step 10.3.5

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

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

Step 10.3.6

Combine the numerators over the common denominator.

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

Step 10.3.7

Multiply $4$ by $- 1$ .

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Cancel the common factor of $4$ .

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Step 12.2.1

Factor $4$ out of $- 4$ .

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

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

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

Step 12.3

Cancel the common factor of $4$ .

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Step 12.3.1

Move the leading negative in $- \frac{\cos (2 x)}{8}$ into the numerator.

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

Step 12.3.2

Factor $4$ out of $- 4$ .

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

Step 12.3.3

Factor $4$ out of $8$ .

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

Step 12.3.4

Cancel the common factor.

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

Step 12.3.5

Rewrite the expression.

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

Step 12.4

Simplify each term.

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Step 12.4.1

Move the negative in front of the fraction.

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

Step 12.4.2

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

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Step 12.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 12.4.2.2

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

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

Step 13

Reorder terms.

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