What is the integral of (2sin(x))/(sqrt(cos^4(x))) ?

The integral of (2sin(x))/(sqrt(cos^4(x))) is 2sec(x)+C

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integral of (2sin(x))/(sqrt(cos^4(x)))

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

2 sec (x) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{sin ( x )}{cos ^{4} ( x )} d x

cos^{4} (x) = cos^{2} (x),

assuming

cos (x) \geq 0

= 2 \cdot \int \frac{sin ( x )}{cos ^{2} ( x )} d x

Apply u-substitution

: \int - \frac{1}{u ^{2}} d u

= 2 \cdot \int - \frac{1}{u ^{2}} d u

Take the constant out:

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

= 2 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

: - \frac{1}{u}

= 2 (- (- \frac{1}{u}))

Substitute back

u = cos (x)

= 2 (- (- \frac{1}{cos ( x )}))

Simplify

2 (- (- \frac{1}{cos ( x )})) : 2 sec (x)

= 2 sec (x)

Add a constant to the solution

= 2 sec (x) + C

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

x \to 3 lim (\frac{4 x + 8}{x - 5})

y = e^{x}, at x = 0

\frac{d}{d x} (\frac{200}{1 + e ^{5 - x}})

\int \frac{12 x}{( x - 1 ) ( x ^{2} + 2 x + 3 )} d x

What is the integral of (2sin(x))/(sqrt(cos^4(x))) ?
The integral of (2sin(x))/(sqrt(cos^4(x))) is 2sec(x)+C