What is the integral of 24x^2arccot(2x^3) ?

The integral of 24x^2arccot(2x^3) is 24(1/3 x^3arccot(2x^3)+1/12 ln|4x^6+1|)+C

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integral of 24x^2arccot(2x^3)

\int 24 x^{2} arccot (2 x^{3}) d x

24 (\frac{1}{3} x^{3} arccot (2 x^{3}) + \frac{1}{12} ln 4 x^{6} + 1) + C

Solution steps

\int 24 x^{2} arccot (2 x^{3}) d x

Take the constant out:

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

= 24 \cdot \int x^{2} arccot (2 x^{3}) d x

Apply Integration By Parts

: \frac{1}{3} x^{3} arccot (2 x^{3}) - \int - \frac{2 x ^{5}}{4 x ^{6} + 1} d x

= 24 (\frac{1}{3} x^{3} arccot (2 x^{3}) - \int - \frac{2 x ^{5}}{4 x ^{6} + 1} d x)

\int - \frac{2 x ^{5}}{4 x ^{6} + 1} d x = - \frac{1}{12} ln 4 x^{6} + 1

= 24 (\frac{1}{3} x^{3} arccot (2 x^{3}) - (- \frac{1}{12} ln 4 x^{6} + 1))

Simplify

= 24 (\frac{1}{3} x^{3} arccot (2 x^{3}) + \frac{1}{12} ln 4 x^{6} + 1)

Add a constant to the solution

= 24 (\frac{1}{3} x^{3} arccot (2 x^{3}) + \frac{1}{12} ln 4 x^{6} + 1) + C

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What is the integral of 24x^2arccot(2x^3) ?
The integral of 24x^2arccot(2x^3) is 24(1/3 x^3arccot(2x^3)+1/12 ln|4x^6+1|)+C