Evaluate the integral.
-$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x$
A) 1 + ln 2
B) $\frac { \ln 3 } { 2 }$
C) -ln 2
D) ln 4

Question

Evaluate the integral. 
-$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x$
A) 1 + ln 2
B) $\frac { \ln 3 } { 2 }$
C) -ln 2
D) ln 4

Quizplus · Accepted Answer

To solve the integral $\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x$, we can use a substitution method. Let $u = 2x + 1$, then $du = 2dx$, or $dx = \frac{du}{2}$. When $x = 1$, $u = 3$, and when $x = 4$, $u = 9$. Substituting these into the integral gives $\int _ { 3 } ^ { 9 } \frac { 1 } { 2 } \cdot \frac { 1 } { u } du = \frac { 1 } { 2 } \ln |u| \Big|_3^9 = \frac { 1 } { 2 } (\ln 9 - \ln 3) = \frac { \ln 3 } { 2 }$, since $\ln 9 = 2\ln 3$.

Evaluate the Integral ∫1412x+1dx\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x∫14​2x+11​dx

Evaluate the Integral $\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x$