What is the integral of (5-50x)/(sqrt(64-25x^2)) ?

The integral of (5-50x)/(sqrt(64-25x^2)) is 2sqrt(-25x^2+64)+arcsin(5/8 x)+C

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integral of (5-50x)/(sqrt(64-25x^2))

\int \frac{5 - 50 x}{64 - 25 x ^{2}} d x

2 - 25 x^{2} + 64 + arcsin (\frac{5}{8} x) + C

Solution steps

\int \frac{5 - 50 x}{64 - 25 x ^{2}} d x

Apply Trigonometric Substitution

: \int - 16 sin (u) + 1 d u

= \int - 16 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 16 sin (u) d u + \int 1 d u

\int 16 sin (u) d u = - 16 cos (u)

\int 1 d u = u

= - (- 16 cos (u)) + u

Substitute back

u = arcsin (\frac{5}{8} x)

= - (- 16 cos (arcsin (\frac{5}{8} x))) + arcsin (\frac{5}{8} x)

Simplify

- (- 16 cos (arcsin (\frac{5}{8} x))) + arcsin (\frac{5}{8} x) : 2 - 25 x^{2} + 64 + arcsin (\frac{5}{8} x)

= 2 - 25 x^{2} + 64 + arcsin (\frac{5}{8} x)

Add a constant to the solution

= 2 - 25 x^{2} + 64 + arcsin (\frac{5}{8} x) + C

x \to 4 + lim (e^{\frac{7}{( 4 - x )}})

4 sin (x) cos (x)

- x^{2} + 7 x, x^{2} - 3 x

\int - x cos (n x) d x

\frac{d}{d x} (cos (3 x))

What is the integral of (5-50x)/(sqrt(64-25x^2)) ?
The integral of (5-50x)/(sqrt(64-25x^2)) is 2sqrt(-25x^2+64)+arcsin(5/8 x)+C