What is the integral of 5arctan(5y) ?

The integral of 5arctan(5y) is 5(yarctan(5y)-1/10 ln|25y^2+1|)+C

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integral of 5arctan(5y)

\int 5 arctan (5 y) d y

5 (y arctan (5 y) - \frac{1}{10} ln 25 y^{2} + 1) + C

Solution steps

\int 5 arctan (5 y) d y

Take the constant out:

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

= 5 \cdot \int arctan (5 y) d y

Apply Integration By Parts

= 5 (y arctan (5 y) - \int \frac{5 y}{25 y ^{2} + 1} d y)

\int \frac{5 y}{25 y ^{2} + 1} d y = \frac{1}{10} ln 25 y^{2} + 1

= 5 (y arctan (5 y) - \frac{1}{10} ln 25 y^{2} + 1)

Add a constant to the solution

= 5 (y arctan (5 y) - \frac{1}{10} ln 25 y^{2} + 1) + C

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What is the integral of 5arctan(5y) ?
The integral of 5arctan(5y) is 5(yarctan(5y)-1/10 ln|25y^2+1|)+C