Calculus Examples

$\int \frac{- 1}{4 x \sqrt{4 - {(2 x)}^{2}}} d x$

Step 1

Simplify.

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Step 1.1

Factor $2$ out of $2 x$ .

$\int \frac{- 1}{4 x \sqrt{4 - {(2 (x))}^{2}}} d x$

Step 1.2

Apply the product rule to $2 (x)$ .

$\int \frac{- 1}{4 x \sqrt{4 - (2^{2} x^{2})}} d x$

Step 1.3

Raise $2$ to the power of $2$ .

$\int \frac{- 1}{4 x \sqrt{4 - (4 x^{2})}} d x$

Step 1.4

Multiply $4$ by $- 1$ .

$\int \frac{- 1}{4 x \sqrt{4 - 4 x^{2}}} d x$

Step 1.5

Move the negative in front of the fraction.

$\int - \frac{1}{4 x \sqrt{4 - 4 x^{2}}} d x$

Step 2

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

$- \int \frac{1}{4 x \sqrt{4 - 4 x^{2}}} d x$

Step 3

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

$- (\frac{1}{4} \int \frac{1}{x \sqrt{4 - 4 x^{2}}} d x)$

Step 4

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{4 - 4 \sin^{2} (t)}} \cos (t) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{4 - 4 \sin^{2} (t)}$ .

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Step 5.1.1

Factor $4$ out of $4$ .

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{4 (1) - 4 \sin^{2} (t)}} \cos (t) d t$

Step 5.1.2

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{4 (1) + 4 (- \sin^{2} (t))}} \cos (t) d t$

Step 5.1.3

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{4 (1 - \sin^{2} (t))}} \cos (t) d t$

Step 5.1.4

Apply pythagorean identity.

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{4 \cos^{2} (t)}} \cos (t) d t$

Step 5.1.5

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

$- \frac{1}{4} \int \frac{1}{\sin (t) \sqrt{{(2 \cos (t))}^{2}}} \cos (t) d t$

Step 5.1.6

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{1}{4} \int \frac{1}{\sin (t) (2 \cos (t))} \cos (t) d t$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $\cos (t)$ .

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Step 5.2.1.1

Factor $\cos (t)$ out of $\sin (t) (2 \cos (t))$ .

$- \frac{1}{4} \int \frac{1}{\cos (t) (\sin (t) \cdot (2))} \cos (t) d t$

Step 5.2.1.2

Cancel the common factor.

$- \frac{1}{4} \int \frac{1}{\cos (t) (\sin (t) \cdot (2))} \cos (t) d t$

Step 5.2.1.3

Rewrite the expression.

$- \frac{1}{4} \int \frac{1}{\sin (t) \cdot (2)} d t$

Step 5.2.2

Move $2$ to the left of $\sin (t)$ .

$- \frac{1}{4} \int \frac{1}{2 \sin (t)} d t$

Step 6

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

$- \frac{1}{4} (\frac{1}{2} \int \frac{1}{\sin (t)} d t)$

Step 7

Simplify.

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Step 7.1

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

$- \frac{1}{4} (\frac{1}{2} \int \csc (t) d t)$

Step 7.2

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

$- \frac{1}{2 \cdot 4} \int \csc (t) d t$

Step 7.3

Multiply $2$ by $4$ .

$- \frac{1}{8} \int \csc (t) d t$

Step 8

The integral of $\csc (t)$ with respect to $t$ is $\ln (| \csc (t) - \cot (t) |)$ .

$- \frac{1}{8} (\ln (| \csc (t) - \cot (t) |) + C)$

Step 9

Simplify.

$- \frac{1}{8} \ln (| \csc (t) - \cot (t) |) + C$

Step 10

Replace all occurrences of $t$ with $\arcsin (x)$ .

$- \frac{1}{8} \ln (| \csc (\arcsin (x)) - \cot (\arcsin (x)) |) + C$