What is the integral of (sec^3(x))/(tan(x)) ?

The integral of (sec^3(x))/(tan(x)) is sec(x)+ln|tan(x/2)|+C

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integral of (sec^3(x))/(tan(x))

\int \frac{sec ^{3} ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{sec ^{3} ( x )}{tan ( x )} d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= \int sec^{3} (x) cot (x) d x

Rewrite using trig identities

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

Apply Integration By Parts

: sec (x) - \int - cot (x) csc (x) tan (x) d x

= sec (x) - \int - cot (x) csc (x) tan (x) d x

\int - cot (x) csc (x) tan (x) d x = - ln tan (\frac{x}{2})

= sec (x) - (- ln tan (\frac{x}{2}))

Simplify

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

Add a constant to the solution

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

x \to e lim (ln ∣ 3 x ∣)

25 y^{^{''}} - y = x e^{\frac{x}{5}}

1 e^{x}

y d x + (y cos (\frac{x}{y}) - x) d y = 0

x \to 4 lim (3 x + 2)

What is the integral of (sec^3(x))/(tan(x)) ?
The integral of (sec^3(x))/(tan(x)) is sec(x)+ln|tan(x/2)|+C