What is the integral of (sec^2(2t))/(sqrt(5+3tan(2t))) ?

The integral of (sec^2(2t))/(sqrt(5+3tan(2t))) is 1/3 sqrt(5+3tan(2t))+C

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integral of (sec^2(2t))/(sqrt(5+3tan(2t)))

\int \frac{sec ^{2} ( 2 t )}{5 + 3 tan ( 2 t )} d t

\frac{1}{3} 5 + 3 tan (2 t) + C

Solution steps

\int \frac{sec ^{2} ( 2 t )}{5 + 3 tan ( 2 t )} d t

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{6} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 5 + 3 tan (2 t)

= \frac{1}{6} \cdot 2 (5 + 3 tan (2 t))^{\frac{1}{2}}

Simplify

\frac{1}{6} \cdot 2 (5 + 3 tan (2 t))^{\frac{1}{2}} : \frac{1}{3} 5 + 3 tan (2 t)

= \frac{1}{3} 5 + 3 tan (2 t)

Add a constant to the solution

= \frac{1}{3} 5 + 3 tan (2 t) + C

\frac{d}{d x} (\frac{x ^{2} - 3 x}{2})

f (x) = 3 x^{2} - x^{3}, at x = 1

\frac{d}{d x} (- 3 sin (x) - 2)

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

f (x) = 11 + 9 x - x^{3}, at x = 1

What is the integral of (sec^2(2t))/(sqrt(5+3tan(2t))) ?
The integral of (sec^2(2t))/(sqrt(5+3tan(2t))) is 1/3 sqrt(5+3tan(2t))+C