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integral of 2/(sqrt(1-(pix)^2))

Solution

\int \frac{2}{1 - ( π x ) ^{2}} d x

1.27323 \dots (\frac{3.14159 \dots x}{4 - 39.47841 \dots x ^{2}} - 0.5 (- arcsin (3.14159 \dots x) + \frac{3.14159 \dots x}{1 - 9.86960 \dots x ^{2}})) + C

Solution steps

\int \frac{2}{1 - ( π x ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{1 - ( π x ) ^{2}} d x

Simplify

(π x)^{2} : 9.86960 \dots x^{2}

= 2 \cdot \int \frac{1}{1 - 9.86960 \dots x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{0.31830 \dots cos ( u )}{1 - 1 sin ^{2} ( u )} d u

= 2 \cdot \int \frac{0.31830 \dots cos ( u )}{1 - 1 sin ^{2} ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot 0.31830 \dots \cdot \int \frac{cos ( u )}{1 - 1 sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{1 - 1 v ^{2}} d v

= 2 \cdot 0.31830 \dots \cdot \int \frac{1}{1 - 1 v ^{2}} d v

Apply Integral Substitution

= 2 \cdot 0.31830 \dots \cdot \int \frac{1}{4 - 1 w ^{2}} d w

Apply Trigonometric Substitution

: \int \frac{2 cos ( t )}{4 - 4 sin ^{2} ( t )} d t

= 2 \cdot 0.31830 \dots \cdot \int \frac{2 cos ( t )}{4 - 4 sin ^{2} ( t )} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot 0.31830 \dots \cdot 2 \cdot \int \frac{cos ( t )}{4 - 4 sin ^{2} ( t )} d t

Apply u-substitution

: \int \frac{1}{4 - 4 r ^{2}} d r

= 2 \cdot 0.31830 \dots \cdot 2 \cdot \int \frac{1}{4 - 4 r ^{2}} d r

Apply Integration By Parts

: \frac{r}{4 - 4 r ^{2}} - \int \frac{4 r ^{2}}{( 4 - 4 r ^{2} ) ^{\frac{3}{2}}} d r

= 2 \cdot 0.31830 \dots \cdot 2 (\frac{r}{4 - 4 r ^{2}} - \int \frac{4 r ^{2}}{( 4 - 4 r ^{2} ) ^{\frac{3}{2}}} d r)

\int \frac{4 r ^{2}}{( 4 - 4 r ^{2} ) ^{\frac{3}{2}}} d r = \frac{1}{2} (- arcsin (r) + \frac{r}{1 - r ^{2}})

= 2 \cdot 0.31830 \dots \cdot 2 (\frac{r}{4 - 4 r ^{2}} - \frac{1}{2} (- arcsin (r) + \frac{r}{1 - r ^{2}}))

Substitute back

= 2 \cdot 0.31830 \dots \cdot 2 \frac{sin ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )}{4 - 4 sin ^{2} ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )} - \frac{1}{2} - arcsin (sin (arcsin (\frac{1}{2} \cdot 2 sin (arcsin (3.14159 \dots x))))) + \frac{sin ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )}{1 - sin ^{2} ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )}

Simplify

2 \cdot 0.31830 \dots \cdot 2 \frac{sin ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )}{4 - 4 sin ^{2} ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )} - \frac{1}{2} - arcsin (sin (arcsin (\frac{1}{2} \cdot 2 sin (arcsin (3.14159 \dots x))))) + \frac{sin ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )}{1 - sin ^{2} ( arcsin ( \frac{1}{2} \cdot 2 sin ( arcsin ( 3.14159 \dots x ) ) ) )} : 1.27323 \dots (\frac{3.14159 \dots x}{4 - 39.47841 \dots x ^{2}} - 0.5 (- arcsin (3.14159 \dots x) + \frac{3.14159 \dots x}{1 - 9.86960 \dots x ^{2}}))

= 1.27323 \dots (\frac{3.14159 \dots x}{4 - 39.47841 \dots x ^{2}} - 0.5 (- arcsin (3.14159 \dots x) + \frac{3.14159 \dots x}{1 - 9.86960 \dots x ^{2}}))

Add a constant to the solution

= 1.27323 \dots (\frac{3.14159 \dots x}{4 - 39.47841 \dots x ^{2}} - 0.5 (- arcsin (3.14159 \dots x) + \frac{3.14159 \dots x}{1 - 9.86960 \dots x ^{2}})) + C

\frac{d}{d x} (f (x) f (x) (x^{2}))

x \to 2 lim (\frac{x - 4}{x - 2})

x \to - \infty lim (x^{3} + x)

\frac{d}{d x} (\frac{tan ( x )}{cos ( x ) - 4})

n = 1 \sum \infty - \frac{5}{2 ^{n - 1}}