What is the integral of (9-162x)/(sqrt(49-81x^2)) ?

The integral of (9-162x)/(sqrt(49-81x^2)) is 2sqrt(-81x^2+49)+arcsin(9/7 x)+C

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integral of (9-162x)/(sqrt(49-81x^2))

\int \frac{9 - 162 x}{49 - 81 x ^{2}} d x

2 - 81 x^{2} + 49 + arcsin (\frac{9}{7} x) + C

Solution steps

\int \frac{9 - 162 x}{49 - 81 x ^{2}} d x

Apply Trigonometric Substitution

: \int - 14 sin (u) + 1 d u

= \int - 14 sin (u) + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 14 sin (u) d u + \int 1 d u

\int 14 sin (u) d u = - 14 cos (u)

\int 1 d u = u

= - (- 14 cos (u)) + u

Substitute back

u = arcsin (\frac{9}{7} x)

= - (- 14 cos (arcsin (\frac{9}{7} x))) + arcsin (\frac{9}{7} x)

Simplify

- (- 14 cos (arcsin (\frac{9}{7} x))) + arcsin (\frac{9}{7} x) : 2 - 81 x^{2} + 49 + arcsin (\frac{9}{7} x)

= 2 - 81 x^{2} + 49 + arcsin (\frac{9}{7} x)

Add a constant to the solution

= 2 - 81 x^{2} + 49 + arcsin (\frac{9}{7} x) + C

y \to 5 + lim (y - 5)

\int csc^{2} (y) cot (y) d y

x \to 3 - lim (\frac{3 - x}{x - 3})

e^{- 3.5 x^{2}}

What is the integral of (9-162x)/(sqrt(49-81x^2)) ?
The integral of (9-162x)/(sqrt(49-81x^2)) is 2sqrt(-81x^2+49)+arcsin(9/7 x)+C