Calculus Examples

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

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

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

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

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

Set up the integral to solve.

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

Use the half-angle formula to rewrite $\cos^{2} (x)$ as $\frac{1 + \cos (2 x)}{2}$ .

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

Use the half-angle formula to rewrite $\sin^{2} (x)$ as $\frac{1 - \cos (2 x)}{2}$ .

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

Simplify.

Tap for more steps...

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

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

Multiply $2$ by $2$ .

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

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

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

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Differentiate $2 x$ .

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

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

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

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

$2 \cdot 1$

Multiply $2$ by $1$ .

$2$

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

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

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

Simplify by multiplying through.

Tap for more steps...

Simplify.

Tap for more steps...

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

$\frac{1}{2 \cdot 4} \int (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Multiply $2$ by $4$ .

$\frac{1}{8} \int (1 - \cos (u_{1})) (1 + \cos (u_{1})) d u_{1}$

Expand $(1 - \cos (u_{1})) (1 + \cos (u_{1}))$ .

Tap for more steps...

Apply the distributive property.

$\frac{1}{8} \int 1 (1 + \cos (u_{1})) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Apply the distributive property.

$\frac{1}{8} \int 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) (1 + \cos (u_{1})) d u_{1}$

Apply the distributive property.

$\frac{1}{8} \int 1 \cdot 1 + 1 \cos (u_{1}) - \cos (u_{1}) \cdot 1 - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Move $\cos (u_{1})$ .

$\frac{1}{8} \int 1 \cdot 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Multiply $1$ by $1$ .

$\frac{1}{8} \int 1 + 1 \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Multiply $\cos (u_{1})$ by $1$ .

$\frac{1}{8} \int 1 + \cos (u_{1}) - 1 \cdot 1 \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Multiply $- 1$ by $1$ .

$\frac{1}{8} \int 1 + \cos (u_{1}) - \cos (u_{1}) - \cos (u_{1}) \cos (u_{1}) d u_{1}$

Factor out negative.

$\frac{1}{8} \int 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos (u_{1}) \cos (u_{1})) d u_{1}$

Raise $\cos (u_{1})$ to the power of $1$ .

$\frac{1}{8} \int 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos (u_{1})) d u_{1}$

Raise $\cos (u_{1})$ to the power of $1$ .

$\frac{1}{8} \int 1 + \cos (u_{1}) - \cos (u_{1}) - (\cos^{1} (u_{1}) \cos^{1} (u_{1})) d u_{1}$

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

$\frac{1}{8} \int 1 + \cos (u_{1}) - \cos (u_{1}) - {\cos (u_{1})}^{1 + 1} d u_{1}$

Add $1$ and $1$ .

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

Subtract $\cos (u_{1})$ from $\cos (u_{1})$ .

$\frac{1}{8} \int 1 + 0 - \cos^{2} (u_{1}) d u_{1}$

Subtract $\cos^{2} (u_{1})$ from $0$ .

$\frac{1}{8} \int 1 - \cos^{2} (u_{1}) d u_{1}$

Split the single integral into multiple integrals.

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

Apply the constant rule.

$\frac{1}{8} (u_{1} + C + \int - \cos^{2} (u_{1}) d u_{1})$

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

$\frac{1}{8} (u_{1} + C - \int \cos^{2} (u_{1}) d u_{1})$

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$\frac{1}{8} (u_{1} + C - \int \frac{1 + \cos (2 u_{1})}{2} d u_{1})$

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

$\frac{1}{8} (u_{1} + C - (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1}))$

Split the single integral into multiple integrals.

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1}))$

Apply the constant rule.

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (u_{1} + C + \int \cos (2 u_{1}) d u_{1}))$

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

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

$2 \cdot 1$

Multiply $2$ by $1$ .

$2$

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

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (u_{1} + C + \int \cos (u_{2}) \frac{1}{2} d u_{2}))$

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

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (u_{1} + C + \int \frac{\cos (u_{2})}{2} d u_{2}))$

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

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (u_{1} + C + \frac{1}{2} \int \cos (u_{2}) d u_{2}))$

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{8} (u_{1} + C - \frac{1}{2} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)))$

Simplify.

Tap for more steps...

Simplify.

$\frac{1}{8} (u_{1} - \frac{u_{1}}{2} - \frac{\sin (u_{2})}{4}) + C$

Simplify.

Tap for more steps...

To write $u_{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Move $2$ to the left of $u_{1}$ .

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

Subtract $u_{1}$ from $2 u_{1}$ .

$\frac{1}{8} (\frac{u_{1}}{2} - \frac{\sin (u_{2})}{4}) + C$

Substitute back in for each integration substitution variable.

Tap for more steps...

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

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

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

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

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

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

Simplify.

Tap for more steps...

Simplify each term.

Tap for more steps...

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Multiply $2$ by $2$ .

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

Apply the distributive property.

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

Combine $\frac{1}{8}$ and $x$ .

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

Multiply $\frac{1}{8} (- \frac{\sin (4 x)}{4})$ .

Tap for more steps...

Multiply $\frac{1}{8}$ by $\frac{\sin (4 x)}{4}$ .

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

Multiply $8$ by $4$ .

$\frac{x}{8} - \frac{\sin (4 x)}{32} + C$

Reorder terms.

$\frac{1}{8} x - \frac{1}{32} \sin (4 x) + C$

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

$F (x) =$ $\frac{1}{8} x - \frac{1}{32} \sin (4 x) + C$