Calculus Examples

${(1 + \sin (x))}^{2}$

Step 1

Write ${(1 + \sin (x))}^{2}$ as a function.

$f (x) = {(1 + \sin (x))}^{2}$

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.

$F (x) = \int {(1 + \sin (x))}^{2} d x$

Step 4

Expand ${(1 + \sin (x))}^{2}$ .

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Step 4.1

Rewrite ${(1 + \sin (x))}^{2}$ as $(1 + \sin (x)) (1 + \sin (x))$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Reorder $\sin (x)$ and $1$ .

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

Step 4.6

Multiply $1$ by $1$ .

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Add $1$ and $1$ .

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

Step 4.13

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Split the single integral into multiple integrals.

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

Step 12

Apply the constant rule.

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

Step 13

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

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

Step 14

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 14.1

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

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Step 14.1.1

Differentiate $2 x$ .

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

Step 14.1.2

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 14.1.3

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 14.1.4

Multiply $2$ by $1$ .

$2$

Step 14.2

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Simplify.

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

Step 19

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

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

Step 20

Simplify.

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Step 20.1

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

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

Step 20.2

Apply the distributive property.

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

Step 20.3

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

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

Step 20.4

Multiply $\frac{1}{2} (- \frac{\sin (2 x)}{2})$ .

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Step 20.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 x)}{2}$ .

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

Step 20.4.2

Multiply $2$ by $2$ .

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

Step 21

Reorder terms.

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

Step 22

The answer is the antiderivative of the function $f (x) = {(1 + \sin (x))}^{2}$ .

$F (x) =$ $x - 2 \cos (x) + \frac{1}{2} x - \frac{1}{4} \sin (2 x) + C$