Calculus Examples

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

Step 1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.2

Multiply $\int (\frac{x}{2} + \frac{\sin (2 x)}{4}) \sin (x) d x$ by $1$ .

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

Step 4.5.3

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

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (x)$ and $d v = \frac{x}{2} + \frac{\sin (2 x)}{4}$ .

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

Step 6

Combine fractions.

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Step 6.1

Combine $\frac{1}{4}$ and $x^{2}$ .

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

Step 6.2

Combine $\cos (2 x)$ and $\frac{1}{8}$ .

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

Step 6.3

Combine $\frac{1}{4}$ and $x^{2}$ .

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

Step 6.4

Combine $\cos (2 x)$ and $\frac{1}{8}$ .

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

Step 7

Solving for $\int \cos^{3} (x) d x$ , we find that $\int \cos^{3} (x) d x$ = $\frac{\cos (x) (\frac{x}{2} + \frac{\sin (2 x)}{4}) + \sin (x) (\frac{x^{2}}{4} - \frac{\cos (2 x)}{8})}{2}$ .

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the answer.

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Step 8.1.1

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

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

Step 8.1.2

Simplify.

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Step 8.1.2.1

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

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

Step 8.1.2.2

Combine.

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

Step 8.1.2.3

Apply the distributive property.

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

Step 8.1.2.4

Cancel the common factor of $2$ .

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Step 8.1.2.4.1

Factor $2$ out of $4$ .

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

Step 8.1.2.4.2

Cancel the common factor.

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

Step 8.1.2.4.3

Rewrite the expression.

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

Step 8.1.2.5

Cancel the common factor of $4$ .

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Step 8.1.2.5.1

Cancel the common factor.

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

Step 8.1.2.5.2

Rewrite the expression.

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

Step 8.1.2.6

Cancel the common factor of $4$ .

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Step 8.1.2.6.1

Cancel the common factor.

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

Step 8.1.2.6.2

Rewrite the expression.

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

Step 8.1.2.7

Multiply $- 1$ by $4$ .

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

Step 8.1.2.8

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

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

Step 8.1.2.9

Cancel the common factor of $- 4$ and $8$ .

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Step 8.1.2.9.1

Factor $4$ out of $- 4 \cos (2 x) \sin (x)$ .

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

Step 8.1.2.9.2

Cancel the common factors.

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Step 8.1.2.9.2.1

Factor $4$ out of $8$ .

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

Step 8.1.2.9.2.2

Cancel the common factor.

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

Step 8.1.2.9.2.3

Rewrite the expression.

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

Step 8.1.2.10

Move the negative in front of the fraction.

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

Step 8.1.2.11

Multiply $4$ by $2$ .

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

Step 8.1.3

Simplify.

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Step 8.1.3.1

Reorder factors in $\frac{2 \cos (x) x + \cos (x) \sin (2 x) + \sin (x) x^{2} - \frac{\cos (2 x) \sin (x)}{2}}{8}$ .

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

Step 8.1.3.2

Reorder terms.

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

Step 8.2

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

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

Step 8.3

Remove parentheses.

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