What is the integral of sqrt(1+(x^2)/(4-x^2)) ?

The integral of sqrt(1+(x^2)/(4-x^2)) is 2arcsin(1/2 x)+C

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integral of sqrt(1+(x^2)/(4-x^2))

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

2 arcsin (\frac{1}{2} x) + C

Solution steps

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

\frac{x ^{2}}{4 - x ^{2}} = \frac{4}{4 - x ^{2}} - 1

= \int 1 + \frac{4}{4 - x ^{2}} - 1 d x

Simplify

1 + \frac{4}{4 - x ^{2}} - 1 : \frac{2}{- x ^{2} + 4}

= \int \frac{2}{- x ^{2} + 4} 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}{- x ^{2} + 4} d x

Apply Trigonometric Substitution

: \int 1 d u

= 2 \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot u

Substitute back

u = arcsin (\frac{1}{2} x)

= 2 \cdot 1 \cdot arcsin (\frac{1}{2} x)

Simplify

= 2 arcsin (\frac{1}{2} x)

Add a constant to the solution

= 2 arcsin (\frac{1}{2} x) + C

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What is the integral of sqrt(1+(x^2)/(4-x^2)) ?
The integral of sqrt(1+(x^2)/(4-x^2)) is 2arcsin(1/2 x)+C