Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

Multiply $\int 1 - \cos (2 x) d x$ by $1$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

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

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

Step 12

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 12.1

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

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Step 12.1.1

Differentiate $2 x$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Apply the constant rule.

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

Step 17

Simplify.

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Step 17.1

Add $x$ and $x$ .

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

Step 17.2

Simplify.

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

Step 18

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

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

Step 19

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

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