Evaluate the integral.
-$$\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x$$
Give your answer in exact form.
A) $\frac { \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$
B) $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 2 \sqrt { 3 } )$
C) $\frac { 2 } { 9 } + \frac { 1 } { 18 } \ln 4$
D) $\frac { 2 \sqrt { 3 } } { 9 } + \frac { 1 } { 18 } \ln ( 7 + 4 \sqrt { 3 } )$

Question

Quizplus · Accepted Answer

The integral of $\sec^3(9x)$ can be solved using integration techniques involving trigonometric identities and substitution. The result involves a combination of a trigonometric term and a logarithmic term. The correct answer must account for the antiderivative of $\sec^3(9x)$ evaluated from $-\pi/27$ to $\pi/27$, which includes a factor that simplifies to $\frac{2\sqrt{3}}{9}$ for the trigonometric part and $\frac{1}{18}\ln(7+4\sqrt{3})$ for the logarithmic part, matching option D.

Evaluate the Integral ∫−π/27π/27sec⁡39xdx\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x∫−π/27π/27​sec39xdx

Evaluate the Integral $\int _ { - \pi / 27 } ^ { \pi / 27 } \sec ^ { 3 } 9 x d x$