What is the integral of 6arctan(8/w) ?

The integral of 6arctan(8/w) is 6(warctan(8/w)+4ln|64+w^2|)+C

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integral of 6arctan(8/w)

\int 6 arctan (\frac{8}{w}) d w

6 (w arctan (\frac{8}{w}) + 4 ln 64 + w^{2}) + C

Solution steps

\int 6 arctan (\frac{8}{w}) d w

Take the constant out:

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

= 6 \cdot \int arctan (\frac{8}{w}) d w

Apply Integration By Parts

: w arctan (\frac{8}{w}) - \int - \frac{8 w}{64 + w ^{2}} d w

= 6 (w arctan (\frac{8}{w}) - \int - \frac{8 w}{64 + w ^{2}} d w)

\int - \frac{8 w}{64 + w ^{2}} d w = - 4 ln 64 + w^{2}

= 6 (w arctan (\frac{8}{w}) - (- 4 ln 64 + w^{2}))

Simplify

= 6 (w arctan (\frac{8}{w}) + 4 ln 64 + w^{2})

Add a constant to the solution

= 6 (w arctan (\frac{8}{w}) + 4 ln 64 + w^{2}) + C

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What is the integral of 6arctan(8/w) ?
The integral of 6arctan(8/w) is 6(warctan(8/w)+4ln|64+w^2|)+C