What is the integral of atan^2(θ) ?

The integral of atan^2(θ) is a(-θ+tan(θ))+C

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integral of atan^2(θ)

\int a tan^{2} (θ) d θ

a (- θ + tan (θ)) + C

Solution steps

\int a tan^{2} (θ) d θ

Take the constant out:

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

= a \cdot \int tan^{2} (θ) d θ

Rewrite using trig identities

= a \cdot \int - 1 + sec^{2} (θ) d θ

Apply the Sum Rule:

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

= a (- \int 1 d θ + \int sec^{2} (θ) d θ)

\int 1 d θ = θ

\int sec^{2} (θ) d θ = tan (θ)

= a (- θ + tan (θ))

Add a constant to the solution

= a (- θ + tan (θ)) + C

\frac{d}{d x} (4 x + ln (x + 1))

\frac{d y}{d x} = \frac{6 x sec ( \frac{y}{x} ) + y}{x}

x \to 0 + lim ((tan (3 x)) x)

x^{2} - 4 x + 25

\frac{\partial}{\partial x} (830 x - \frac{25000}{x} - \frac{5400}{y} + 2280 y + 457950)