Using the first three nonzero terms of the Maclaurin series, the approximated value of $\int _ { 1 } ^ { 2 } \frac { \sin x ^ { 2 } } { x } d x$ is

A) $\frac { 241 } { 400 }$
B) $\frac { 241 } { 440 }$
C) $\frac { 241 } { 500 }$
D) $\frac { 241 } { 550 }$
E) $\frac { 241 } { 600 }$

Question

Using the first three nonzero terms of the Maclaurin series, the approximated value of $\int _ { 1 } ^ { 2 } \frac { \sin x ^ { 2 } } { x } d x$ is

A) $\frac { 241 } { 400 }$
B) $\frac { 241 } { 440 }$
C) $\frac { 241 } { 500 }$
D) $\frac { 241 } { 550 }$
E) $\frac { 241 } { 600 }$

Quizplus · Accepted Answer

The Maclaurin series for $\sin x^2$ is $x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} + \cdots$. Dividing by $x$, the series becomes $x - \frac{x^5}{6} + \frac{x^9}{120} + \cdots$. Integrating term by term from 1 to 2 and using only the first three terms gives $\frac{241}{400}$.

Using the First Three Nonzero Terms of the Maclaurin Series ∫12sin⁡x2xdx\int _ { 1 } ^ { 2 } \frac { \sin x ^ { 2 } } { x } d x∫12​xsinx2​dx

Using the First Three Nonzero Terms of the Maclaurin Series $\int _ { 1 } ^ { 2 } \frac { \sin x ^ { 2 } } { x } d x$