What is the integral of (cos(x))/((cos^2(x))^{3/2)} ?

The integral of (cos(x))/((cos^2(x))^{3/2)} is tan(x)+C

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integral of (cos(x))/((cos^2(x))^{3/2)}

\int \frac{cos ( x )}{( cos ^{2} ( x ) ) ^{\frac{3}{2}}} d x

tan (x) + C

Solution steps

\int \frac{cos ( x )}{( cos ^{2} ( x ) ) ^{\frac{3}{2}}} d x

(cos^{2} (x))^{\frac{3}{2}} = cos^{3} (x),

assuming

cos (x) \geq 0

= \int \frac{cos ( x )}{cos ^{3} ( x )} d x

Simplify

\frac{cos ( x )}{cos ^{3} ( x )} : sec^{2} (x)

= \int sec^{2} (x) d x

Use the common integral:

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

= tan (x)

Add a constant to the solution

= tan (x) + C

\int \frac{3 x - 24}{x ^{3} - 4 x ^{2} + 4 x} d x

\int_{0}^{2} e^{4 x} d x

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

x \to - 4 lim (3 ln (9) (7 - 5 x))

x \to 0 lim (sin (x) - e^{x})

What is the integral of (cos(x))/((cos^2(x))^{3/2)} ?
The integral of (cos(x))/((cos^2(x))^{3/2)} is tan(x)+C