Use the substitution formula to evaluate the integral.
-$$\int _ { 0 } ^ { 7 \pi / 4 } \tan \frac { x } { 7 } d x$$
A) $\frac { - 7 \ln 2 } { 2 }$
B) $\frac { 7 \sqrt { 2 } } { 2 }$
C) $\frac { 7 \ln 2 } { 2 }$
D) $\frac { - 7 \sqrt { 2 } } { 2 }$

Question

Quizplus · Accepted Answer

The correct answer is C because the substitution formula for integrals is $\int f(g(x))g'(x)dx = \int f(u)du$, where $u = g(x)$. In this case, let $u = \frac{x}{7}$, so $du = \frac{1}{7}dx$ and $g(x) = 7u$. Substituting into the integral, we get:$$\int_0^{7\pi/4} \tan \frac{x}{7} dx = \int_0^\pi \tan u \cdot \frac{1}{7}du = \frac{1}{7}\int_0^\pi \tan u du = \frac{1}{7}\ln|\sec u|_0^\pi = \frac{1}{7}\ln|\sec \pi - \sec 0| = \frac{1}{7}\ln|\infty - 1| = \frac{1}{7}\ln \infty = \infty$$Therefore, the integral diverges and does not have a finite value.

Use the Substitution Formula to Evaluate the Integral ∫07π/4tan⁡x7dx\int _ { 0 } ^ { 7 \pi / 4 } \tan \frac { x } { 7 } d x∫07π/4​tan7x​dx

Use the Substitution Formula to Evaluate the Integral $\int _ { 0 } ^ { 7 \pi / 4 } \tan \frac { x } { 7 } d x$