What is the integral of 4/((4-(2sin(x))^2)^{3/2)} ?

The integral of 4/((4-(2sin(x))^2)^{3/2)} is 1/2 (1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|)+C

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integral of 4/((4-(2sin(x))^2)^{3/2)}

\int \frac{4}{( 4 - ( 2 sin ( x ) ) ^{2} ) ^{\frac{3}{2}}} d x

\frac{1}{2} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

Solution steps

\int \frac{4}{( 4 - ( 2 sin ( x ) ) ^{2} ) ^{\frac{3}{2}}} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{( 4 - ( 2 sin ( x ) ) ^{2} ) ^{\frac{3}{2}}} d x

Simplify

\frac{1}{( 4 - ( 2 sin ( x ) ) ^{2} ) ^{\frac{3}{2}}} : \frac{1}{8} sec^{3} (x)

= 4 \cdot \int \frac{1}{8} sec^{3} (x) d x

Take the constant out:

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

= 4 \cdot \frac{1}{8} \cdot \int sec^{3} (x) d x

Apply Integral Reduction

= 4 \cdot \frac{1}{8} (\frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} \cdot \int sec (x) d x)

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

= 4 \cdot \frac{1}{8} (\frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Simplify

4 \cdot \frac{1}{8} (\frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) : \frac{1}{2} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

= \frac{1}{2} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Add a constant to the solution

= \frac{1}{2} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

\int \frac{2 x - 4 + 4 x}{x} d x

\int \frac{t}{1 + t ^{3}} d t

\int 5 x e^{- 3 x^{2}} d x

\int \frac{5 x ^{2} + 2}{x ^{3} - 4 x ^{2} + 5 x} d x

\int \frac{d}{1} (x y) d x

What is the integral of 4/((4-(2sin(x))^2)^{3/2)} ?
The integral of 4/((4-(2sin(x))^2)^{3/2)} is 1/2 (1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|)+C