What is the integral of ((2+w))/(sqrt(13+12w-w^2)) ?

The integral of ((2+w))/(sqrt(13+12w-w^2)) is 8arcsin(1/7 (w-6))-sqrt(-w^2+12w+13)+C

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integral of ((2+w))/(sqrt(13+12w-w^2))

\int \frac{( 2 + w )}{13 + 12 w - w ^{2}} d w

8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + C

Solution steps

\int \frac{2 + w}{13 + 12 w - w ^{2}} d w

Expand

\frac{2 + w}{13 + 12 w - w ^{2}} : \frac{2}{13 + 12 w - w ^{2}} + \frac{w}{13 + 12 w - w ^{2}}

= \int \frac{2}{13 + 12 w - w ^{2}} + \frac{w}{13 + 12 w - w ^{2}} d w

Apply the Sum Rule:

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

= \int \frac{2}{13 + 12 w - w ^{2}} d w + \int \frac{w}{13 + 12 w - w ^{2}} d w

\int \frac{2}{13 + 12 w - w ^{2}} d w = 2 arcsin (\frac{1}{7} (w - 6))

\int \frac{w}{13 + 12 w - w ^{2}} d w = - - w^{2} + 12 w + 13 + 6 arcsin (\frac{1}{7} (w - 6))

= 2 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + 6 arcsin (\frac{1}{7} (w - 6))

Simplify

= 8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13

Add a constant to the solution

= 8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + C

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What is the integral of ((2+w))/(sqrt(13+12w-w^2)) ?
The integral of ((2+w))/(sqrt(13+12w-w^2)) is 8arcsin(1/7 (w-6))-sqrt(-w^2+12w+13)+C