Evaluate the improper integral $\int _ { 0 } ^ { 3 x / 4 } \sec ^ { 2 } x d x$
A) $- 1$

B) $\frac { 1 } { 2 }$

C) $\pi$
D)2
E) $\frac { \pi } { 16 }$
F) $\frac { \sqrt { 3 } } { 2 }$
G)1
H)Divergent

Question

Evaluate the improper integral $\int _ { 0 } ^ { 3 x / 4 } \sec ^ { 2 } x d x$ 
A) $- 1$

B) $\frac { 1 } { 2 }$

C) $\pi$ 
D)2
E) $\frac { \pi } { 16 }$ 
F) $\frac { \sqrt { 3 } } { 2 }$ 
G)1
H)Divergent

Quizplus · Accepted Answer

The integral of $\sec^2 x$ is $\tan x$, so the integral from 0 to $3x/4$ does not make sense as written because the upper limit of integration should be a constant value, not a function of $x$. Moreover, $\sec^2 x$ is undefined at $x = \frac{\pi}{2}$ (and every odd multiple thereof), which lies within the interval implied by $0$ to $3x/4$ for some values of $x$. Therefore, without a proper upper limit of integration, this integral cannot be evaluated in the conventional sense and is considered divergent.

Evaluate the Improper Integral ∫03x/4sec⁡2xdx\int _ { 0 } ^ { 3 x / 4 } \sec ^ { 2 } x d x∫03x/4​sec2xdx

Evaluate the Improper Integral $\int _ { 0 } ^ { 3 x / 4 } \sec ^ { 2 } x d x$