What is the integral of sec^{3x}(x ?

The integral of sec^{3x}(x is 1/(9ln^2(sec(d)))(sec^{3x}(d)ln(sec^{3x}(d))-sec^{3x}(d))+C

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integral of sec^{3x}(x

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

\frac{1}{9 ln ^{2} ( sec ( d ) )} (sec^{3 x} (d) ln (sec^{3 x} (d)) - sec^{3 x} (d)) + C

Solution steps

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

Apply u-substitution

: \int \frac{ln ( u )}{9 ln ^{2} ( sec ( d ) )} d u

= \int \frac{ln ( u )}{9 ln ^{2} ( sec ( d ) )} d u

Take the constant out:

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

= \frac{1}{9 ln ^{2} ( sec ( d ) )} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= \frac{1}{9 ln ^{2} ( sec ( d ) )} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{9 ln ^{2} ( sec ( d ) )} (u ln (u) - u)

Substitute back

u = sec^{3 x} (d)

= \frac{1}{9 ln ^{2} ( sec ( d ) )} (sec^{3 x} (d) ln (sec^{3 x} (d)) - sec^{3 x} (d))

Add a constant to the solution

= \frac{1}{9 ln ^{2} ( sec ( d ) )} (sec^{3 x} (d) ln (sec^{3 x} (d)) - sec^{3 x} (d)) + C

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What is the integral of sec^{3x}(x ?
The integral of sec^{3x}(x is 1/(9ln^2(sec(d)))(sec^{3x}(d)ln(sec^{3x}(d))-sec^{3x}(d))+C