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

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

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

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

2 x + 2 x - x^{2} + 1 - \frac{3}{2} sin (2 arcsin (x)) + \frac{1}{2} ln - x^{2} + 1 + 1 - \frac{1}{2} ln - x^{2} + 1 - 1 - - x^{2} + 1 - arcsin (x) + ln ∣ x ∣ - 1 + C

Solution steps

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

Apply Integration By Parts

: - 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - \int - 4 + 4 1 - x^{2} + 4 x arcsin (x) - \frac{1}{x} + \frac{1 - x ^{2}}{x} + arcsin (x) d x

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

\int - 4 + 4 1 - x^{2} + 4 x arcsin (x) - \frac{1}{x} + \frac{1 - x ^{2}}{x} + arcsin (x) d x = 2 x^{2} arcsin (x) - 4 x + x arcsin (x) + \frac{3}{2} sin (2 arcsin (x)) - \frac{1}{2} ln - x^{2} + 1 + 1 + \frac{1}{2} ln - x^{2} + 1 - 1 + arcsin (x) - ln ∣ x ∣ + 2 - x^{2} + 1

= - 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - (2 x^{2} arcsin (x) - 4 x + x arcsin (x) + \frac{3}{2} sin (2 arcsin (x)) - \frac{1}{2} ln - x^{2} + 1 + 1 + \frac{1}{2} ln - x^{2} + 1 - 1 + arcsin (x) - ln ∣ x ∣ + 2 - x^{2} + 1)

Simplify

- 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - (2 x^{2} arcsin (x) - 4 x + x arcsin (x) + \frac{3}{2} sin (2 arcsin (x)) - \frac{1}{2} ln - x^{2} + 1 + 1 + \frac{1}{2} ln - x^{2} + 1 - 1 + arcsin (x) - ln ∣ x ∣ + 2 - x^{2} + 1) : 2 x + 2 x - x^{2} + 1 - \frac{3}{2} sin (2 arcsin (x)) + \frac{1}{2} ln - x^{2} + 1 + 1 - \frac{1}{2} ln - x^{2} + 1 - 1 - - x^{2} + 1 - arcsin (x) + ln ∣ x ∣ - 1

= 2 x + 2 x - x^{2} + 1 - \frac{3}{2} sin (2 arcsin (x)) + \frac{1}{2} ln - x^{2} + 1 + 1 - \frac{1}{2} ln - x^{2} + 1 - 1 - - x^{2} + 1 - arcsin (x) + ln ∣ x ∣ - 1

Add a constant to the solution

= 2 x + 2 x - x^{2} + 1 - \frac{3}{2} sin (2 arcsin (x)) + \frac{1}{2} ln - x^{2} + 1 + 1 - \frac{1}{2} ln - x^{2} + 1 - 1 - - x^{2} + 1 - arcsin (x) + ln ∣ x ∣ - 1 + C

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

\int x (x^{2} - 5) d x

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

\int e^{2 x - 1} d x

\int v^{2} e^{- v^{2}} d v

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