What is the integral of 5arctan(2y) ?

The integral of 5arctan(2y) is 5(yarctan(2y)-1/4 ln|4y^2+1|)+C

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integral of 5arctan(2y)

\int 5 arctan (2 y) d y

5 (y arctan (2 y) - \frac{1}{4} ln 4 y^{2} + 1) + C

Solution steps

\int 5 arctan (2 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 (2 y) d y

Apply Integration By Parts

: y arctan (2 y) - \int \frac{2 y}{4 y ^{2} + 1} d y

= 5 (y arctan (2 y) - \int \frac{2 y}{4 y ^{2} + 1} d y)

\int \frac{2 y}{4 y ^{2} + 1} d y = \frac{1}{4} ln 4 y^{2} + 1

= 5 (y arctan (2 y) - \frac{1}{4} ln 4 y^{2} + 1)

Add a constant to the solution

= 5 (y arctan (2 y) - \frac{1}{4} ln 4 y^{2} + 1) + C

\frac{\partial}{\partial x} (ln (2 + x^{2} + 2 y^{2}))

\int - \frac{1}{4} d x

\int t \cdot sin (t^{2}) d t

\frac{d}{d x} (- sin (x - 2))

What is the integral of 5arctan(2y) ?
The integral of 5arctan(2y) is 5(yarctan(2y)-1/4 ln|4y^2+1|)+C