What is the integral of 2/(sqrt(11+10w-w^2)) ?

The integral of 2/(sqrt(11+10w-w^2)) is 2arcsin(1/6 (w-5))+C

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integral of 2/(sqrt(11+10w-w^2))

\int \frac{2}{11 + 10 w - w ^{2}} d w

2 arcsin (\frac{1}{6} (w - 5)) + C

Solution steps

\int \frac{2}{11 + 10 w - w ^{2}} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{11 + 10 w - w ^{2}} d w

Complete the square

11 + 10 w - w^{2} : - (w - 5)^{2} + 36

= 2 \cdot \int \frac{1}{- ( w - 5 ) ^{2} + 36} d w

Apply u-substitution

: \int \frac{1}{- u ^{2} + 36} d u

= 2 \cdot \int \frac{1}{- u ^{2} + 36} d u

Apply Trigonometric Substitution

: \int 1 d v

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

= 2 \cdot 1 \cdot arcsin (\frac{1}{6} (w - 5))

Simplify

= 2 arcsin (\frac{1}{6} (w - 5))

Add a constant to the solution

= 2 arcsin (\frac{1}{6} (w - 5)) + C

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What is the integral of 2/(sqrt(11+10w-w^2)) ?
The integral of 2/(sqrt(11+10w-w^2)) is 2arcsin(1/6 (w-5))+C