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

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

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Step 4.3.1.2.1

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 4.3.1.2.2

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 4.3.1.2.3

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 4.3.1.2.4

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

Add $\cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

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

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 4.6

Simplify each term.

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Step 4.6.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 4.6.2

Reorder $\sin (x)$ and $2$ .

$\int 1 + 2 \cdot \sin (x) \cos (x) d x$

Step 4.6.3

Apply the sine double-angle identity.

$\int 1 + \sin (2 x) d x$

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

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

Step 8

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

$x + C + \int \frac{\sin (u)}{2} d u$

Step 9

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

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

Step 10

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

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

Step 11

Simplify.

$x - \frac{\cos (u)}{2} + C$

Step 12

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

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

Step 13

Reorder terms.

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

Step 14

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

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