What is the integral of xcsc((pi-x^2)/3) ?

The integral of xcsc((pi-x^2)/3) is -3/2 ln|tan(1/6 (pi-x^2))|+C

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integral of xcsc((pi-x^2)/3)

\int x csc (\frac{π - x ^{2}}{3}) d x

- \frac{3}{2} ln tan (\frac{1}{6} (π - x^{2})) + C

Solution steps

\int x csc (\frac{π - x ^{2}}{3}) d x

Apply u-substitution

: \int - \frac{csc ( \frac{u}{3} )}{2} d u

= \int - \frac{csc ( \frac{u}{3} )}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int csc (\frac{u}{3}) d u

Apply u-substitution

: \int csc (v) \cdot 3 d v

= - \frac{1}{2} \cdot \int csc (v) \cdot 3 d v

Take the constant out:

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

= - \frac{1}{2} \cdot 3 \cdot \int csc (v) d v

Apply u-substitution

: \int \frac{1}{w} d w

= - \frac{1}{2} \cdot 3 \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= - \frac{1}{2} \cdot 3 ln ∣ w ∣

Substitute back

= - \frac{1}{2} \cdot 3 ln tan (\frac{\frac{π - x ^{2}}{3}}{2})

Simplify

- \frac{1}{2} \cdot 3 ln tan (\frac{\frac{π - x ^{2}}{3}}{2}) : - \frac{3}{2} ln tan (\frac{1}{6} (π - x^{2}))

= - \frac{3}{2} ln tan (\frac{1}{6} (π - x^{2}))

Add a constant to the solution

= - \frac{3}{2} ln tan (\frac{1}{6} (π - x^{2})) + C

f (x) = 4 arcsin (x^{4})

\frac{d}{d x} (- x^{5} - \frac{x ^{3}}{6} + \frac{3}{x} - \frac{1}{x ^{2}})

y = \frac{7}{( 6 x ) ^{6}}

4 x, 64 - x^{2}, [0, 6.928]

x \to 69.9 lim (\frac{ln ( \frac{x - 70}{230} )}{- 0.19})

What is the integral of xcsc((pi-x^2)/3) ?
The integral of xcsc((pi-x^2)/3) is -3/2 ln|tan(1/6 (pi-x^2))|+C