Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Expand $(\sin (2 x) - \cos (2 x)) (\sin (2 x) - \cos (2 x))$ using the FOIL Method.

Tap for more steps...

Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

Tap for more steps...

Step 4.3.1

Simplify each term.

Tap for more steps...

Step 4.3.1.1

Multiply $\sin (2 x) \sin (2 x)$ .

Tap for more steps...

Step 4.3.1.1.1

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 4.3.1.1.2

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 4.3.1.1.3

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

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

Step 4.3.1.1.4

Add $1$ and $1$ .

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

Step 4.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3

Multiply $- \cos (2 x) (- \cos (2 x))$ .

Tap for more steps...

Step 4.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.3.2

Multiply $\cos (2 x)$ by $1$ .

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

Step 4.3.1.3.3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 4.3.1.3.4

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 4.3.1.3.5

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

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

Step 4.3.1.3.6

Add $1$ and $1$ .

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

Step 4.3.2

Reorder the factors of $- \sin (2 x) \cos (2 x)$ .

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

Step 4.3.3

Subtract $\cos (2 x) \sin (2 x)$ from $- \cos (2 x) \sin (2 x)$ .

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

Step 4.4

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

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

Step 4.5

Apply pythagorean identity.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

Let $u_{2} = \cos (2 x)$ . Then $d u_{2} = - 2 \sin (2 x) d x$ , so $- \frac{1}{2} d u_{2} = \sin (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 8.1

Let $u_{2} = \cos (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 8.1.1

Differentiate $\cos (2 x)$ .

$\frac{d}{d x} [\cos (2 x)]$

Step 8.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

Tap for more steps...

Step 8.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]$

Step 8.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d x} [2 x]$

Step 8.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- \sin (2 x) \frac{d}{d x} [2 x]$

Step 8.1.3

Differentiate.

Tap for more steps...

Step 8.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- \sin (2 x) (2 \frac{d}{d x} [x])$

Step 8.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 x) \frac{d}{d x} [x]$

Step 8.1.3.3

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

$- 2 \sin (2 x) \cdot 1$

Step 8.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 x)$

Step 8.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$x + C - 2 \int u_{2} \frac{1}{- 2} d u_{2}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Move the negative in front of the fraction.

$x + C - 2 \int u_{2} (- \frac{1}{2}) d u_{2}$

Step 9.2

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

$x + C - 2 \int - \frac{u_{2}}{2} d u_{2}$

Step 10

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$x + C - 2 (- \int \frac{u_{2}}{2} d u_{2})$

Step 11

Multiply $- 1$ by $- 2$ .

$x + C + 2 \int \frac{u_{2}}{2} d u_{2}$

Step 12

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

$x + C + 2 (\frac{1}{2} \int u_{2} d u_{2})$

Step 13

Simplify.

Tap for more steps...

Step 13.1

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

$x + C + \frac{2}{2} \int u_{2} d u_{2}$

Step 13.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.2.1

Cancel the common factor.

$x + C + \frac{2}{2} \int u_{2} d u_{2}$

Step 13.2.2

Rewrite the expression.

$x + C + 1 \int u_{2} d u_{2}$

Step 13.3

Multiply $\int u_{2} d u_{2}$ by $1$ .

$x + C + \int u_{2} d u_{2}$

Step 14

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

$x + C + \frac{1}{2} {u_{2}}^{2} + C$

Step 15

Simplify.

$x + \frac{1}{2} {u_{2}}^{2} + C$

Step 16

Replace all occurrences of $u_{2}$ with $\cos (2 x)$ .

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

Step 17

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

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