Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
-$$\int _ { 0 } ^ { 1 } 5 \sqrt { x ^ { 2 } + 1 } d x$$
A) $\frac { 5 } { 2 } [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$
B) $5 [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$
C) $\sqrt { 2 } + \ln ( \sqrt { 2 } + 1 )$
D) $5 [ \sqrt { 2 } - \ln ( \sqrt { 2 } + 1 ) ]$

Question

Quizplus · Accepted Answer

The correct substitution for this integral involves setting $x = \tan(\theta)$, which leads to $dx = \sec^2(\theta) d\theta$ and transforms the integral into a form that can be directly integrated using trigonometric identities and the reduction formula. The calculation yields $\frac { 5 } { 2 } [ \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) ]$ as the result.

Evaluate the Integral by Making a Substitution (Possibly Trigonometric) and Then