What is the integral of 1/(sqrt(-5w^2+22w-8)) ?

The integral of 1/(sqrt(-5w^2+22w-8)) is

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integral of 1/(sqrt(-5w^2+22w-8))

\int \frac{1}{- 5 w ^{2} + 22 w - 8} d w

5 (\frac{1}{- 25 w ^{2} + 110 w - 40} w - \frac{11}{5} \frac{1}{- 25 w ^{2} + 110 w - 40} + \frac{1}{5} arcsin (\frac{5}{9} w - \frac{11}{9}) - \frac{5 w - 11}{5 - 25 w ^{2} + 110 w - 40}) + C

Solution steps

\int \frac{1}{- 5 w ^{2} + 22 w - 8} d w

Complete the square

- 5 w^{2} + 22 w - 8 : - 5 (w - \frac{11}{5})^{2} + \frac{81}{5}

= \int \frac{1}{- 5 ( w - \frac{11}{5} ) ^{2} + \frac{81}{5}} d w

Apply u-substitution

: \int \frac{5}{- 25 u ^{2} + 81} d u

= \int \frac{5}{- 25 u ^{2} + 81} d u

\frac{5}{- 25 u ^{2} + 81} = 5 \frac{1}{- 25 u ^{2} + 81}

= \int 5 \frac{1}{- 25 u ^{2} + 81} d u

Take the constant out:

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

= 5 \cdot \int \frac{1}{- 25 u ^{2} + 81} d u

Apply Integration By Parts

: \frac{1}{- 25 u ^{2} + 81} u - \int \frac{25 u ^{2}}{( - 25 u ^{2} + 81 ) ^{\frac{3}{2}}} d u

= 5 (\frac{1}{- 25 u ^{2} + 81} u - \int \frac{25 u ^{2}}{( - 25 u ^{2} + 81 ) ^{\frac{3}{2}}} d u)

\int \frac{25 u ^{2}}{( - 25 u ^{2} + 81 ) ^{\frac{3}{2}}} d u = - \frac{1}{5} arcsin (\frac{5}{9} u) + \frac{u}{81 - 25 u ^{2}}

= 5 (\frac{1}{- 25 u ^{2} + 81} u - (- \frac{1}{5} arcsin (\frac{5}{9} u) + \frac{u}{81 - 25 u ^{2}}))

Substitute back

u = w - \frac{11}{5}

= 5 \frac{1}{- 25 ( w - \frac{11}{5} ) ^{2} + 81} (w - \frac{11}{5}) - - \frac{1}{5} arcsin (\frac{5}{9} (w - \frac{11}{5})) + \frac{w - \frac{11}{5}}{81 - 25 ( w - \frac{11}{5} ) ^{2}}

Simplify

5 \frac{1}{- 25 ( w - \frac{11}{5} ) ^{2} + 81} (w - \frac{11}{5}) - - \frac{1}{5} arcsin (\frac{5}{9} (w - \frac{11}{5})) + \frac{w - \frac{11}{5}}{81 - 25 ( w - \frac{11}{5} ) ^{2}} : 5 (\frac{1}{- 25 w ^{2} + 110 w - 40} w - \frac{11}{5} \frac{1}{- 25 w ^{2} + 110 w - 40} + \frac{1}{5} arcsin (\frac{5}{9} w - \frac{11}{9}) - \frac{5 w - 11}{5 - 25 w ^{2} + 110 w - 40})

= 5 (\frac{1}{- 25 w ^{2} + 110 w - 40} w - \frac{11}{5} \frac{1}{- 25 w ^{2} + 110 w - 40} + \frac{1}{5} arcsin (\frac{5}{9} w - \frac{11}{9}) - \frac{5 w - 11}{5 - 25 w ^{2} + 110 w - 40})

Add a constant to the solution

= 5 (\frac{1}{- 25 w ^{2} + 110 w - 40} w - \frac{11}{5} \frac{1}{- 25 w ^{2} + 110 w - 40} + \frac{1}{5} arcsin (\frac{5}{9} w - \frac{11}{9}) - \frac{5 w - 11}{5 - 25 w ^{2} + 110 w - 40}) + C

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What is the integral of 1/(sqrt(-5w^2+22w-8)) ?
The integral of 1/(sqrt(-5w^2+22w-8)) is