What is the integral of sqrt(ln(sec(x)))tan(x) ?

The integral of sqrt(ln(sec(x)))tan(x) is 1/(3sqrt(2))ln^{3/2}(sec^2(x))+C

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integral of sqrt(ln(sec(x)))tan(x)

\int ln (sec (x)) tan (x) d x

\frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x)) + C

Solution steps

\int ln (sec (x)) tan (x) d x

Apply u-substitution

: \int \frac{u ln ( 1 + u ^{2} )}{2 ( 1 + u ^{2} )} d u

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{u ln ( 1 + u ^{2} )}{1 + u ^{2}} d u

Apply u-substitution

: \int v^{2} d v

= \frac{1}{2} \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{2} \cdot \frac{v ^{3}}{3}

Substitute back

= \frac{1}{2} \cdot \frac{( ln ( 1 + tan ^{2} ( x ) ) ) ^{3}}{3}

Simplify

\frac{1}{2} \cdot \frac{( ln ( 1 + tan ^{2} ( x ) ) ) ^{3}}{3} : \frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x))

= \frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x))

Add a constant to the solution

= \frac{1}{3 2} ln^{\frac{3}{2}} (sec^{2} (x)) + C

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