What is the integral of 5tan^2(y) ?

The integral of 5tan^2(y) is 5(-y+tan(y))+C

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integral of 5tan^2(y)

\int 5 tan^{2} (y) d y

5 (- y + tan (y)) + C

Solution steps

\int 5 tan^{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 tan^{2} (y) d y

Rewrite using trig identities

= 5 \cdot \int - 1 + sec^{2} (y) d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (- \int 1 d y + \int sec^{2} (y) d y)

\int 1 d y = y

\int sec^{2} (y) d y = tan (y)

= 5 (- y + tan (y))

Add a constant to the solution

= 5 (- y + tan (y)) + C

\int sin^{2} (\frac{5 x}{2}) d x

(2 x + 1) y^{^{'}} + 2 y = 4 x

\frac{d}{d x} (1 - sec^{2} (x))

\int \frac{1}{5 y} d y

y^{^{''}} + y = 0, y (\frac{π}{3}) = 0, y^{^{'}} (\frac{π}{3}) = 8