Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

$\int (\sin (x) - \cos (x)) (\sin (x) - \cos (x)) d x$

Step 4.2

Expand $(\sin (x) - \cos (x)) (\sin (x) - \cos (x))$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\int \sin (x) (\sin (x) - \cos (x)) - \cos (x) (\sin (x) - \cos (x)) d x$

Step 4.2.2

Apply the distributive property.

$\int \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos (x) (\sin (x) - \cos (x)) d x$

Step 4.2.3

Apply the distributive property.

$\int \sin (x) \sin (x) + \sin (x) (- \cos (x)) - \cos (x) \sin (x) - \cos (x) (- \cos (x)) d x$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $\sin (x) \sin (x)$ .

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Step 4.3.1.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.3.1.1.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.1.1.4

Add $1$ and $1$ .

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

Step 4.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3

Multiply $- \cos (x) (- \cos (x))$ .

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Step 4.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.3.2

Multiply $\cos (x)$ by $1$ .

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

Step 4.3.1.3.3

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.3.1.3.4

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.3.1.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3.1.3.6

Add $1$ and $1$ .

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

Step 4.3.2

Reorder the factors of $- \sin (x) \cos (x)$ .

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

Step 4.3.3

Subtract $\cos (x) \sin (x)$ from $- \cos (x) \sin (x)$ .

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

Step 4.4

Move $\cos^{2} (x)$ .

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

Step 4.5

Apply pythagorean identity.

$\int 1 - 2 \cos (x) \sin (x) d x$

Step 5

Split the single integral into multiple integrals.

$\int d x + \int - 2 \cos (x) \sin (x) d x$

Step 6

Apply the constant rule.

$x + C + \int - 2 \cos (x) \sin (x) d x$

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$x + C - 2 \int \cos (x) \sin (x) d x$

Step 8

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- \frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $\cos (x)$ .

$\frac{d}{d x} [\cos (x)]$

Step 8.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x)$

Step 8.2

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

$x + C - 2 \int - u d u$

Step 9

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

$x + C - 2 (- \int u d u)$

Step 10

Multiply $- 1$ by $- 2$ .

$x + C + 2 \int u d u$

Step 11

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

$x + C + 2 (\frac{1}{2} u^{2} + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

$x + 2 (\frac{1}{2}) u^{2} + C$

Step 12.2

Simplify.

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Step 12.2.1

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

$x + \frac{2}{2} u^{2} + C$

Step 12.2.2

Cancel the common factor of $2$ .

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Step 12.2.2.1

Cancel the common factor.

$x + \frac{2}{2} u^{2} + C$

Step 12.2.2.2

Rewrite the expression.

$x + 1 u^{2} + C$

Step 12.2.3

Multiply $u^{2}$ by $1$ .

$x + u^{2} + C$

Step 13

Replace all occurrences of $u$ with $\cos (x)$ .

$x + \cos^{2} (x) + C$

Step 14

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

$F (x) =$ $x + \cos^{2} (x) + C$