Find the coefficient of $x ^ { 5 }$ in the Maclaurin series for $f ( x ) = \int \cos \left( x ^ { 2 } \right) d x$ .Note: The series is unique except for the constant of integration.
A) $- \frac { 1 } { 10 }$
B) $\frac { 1 } { 15 }$
C) $- \frac { 1 } { 5 }$
D) $\frac { 2 } { 5 }$
E) $- \frac { 2 } { 5 }$
F) $- \frac { 1 } { 15 }$
G) $\frac { 1 } { 5 }$
H) $\frac { 1 } { 10 }$

Question

Find the coefficient of $x ^ { 5 }$ in the Maclaurin series for $f ( x ) = \int \cos \left( x ^ { 2 } \right) d x$ .Note: The series is unique except for the constant of integration.
A) $- \frac { 1 } { 10 }$ 
B) $\frac { 1 } { 15 }$ 
C) $- \frac { 1 } { 5 }$ 
D) $\frac { 2 } { 5 }$ 
E) $- \frac { 2 } { 5 }$ 
F) $- \frac { 1 } { 15 }$ 
G) $\frac { 1 } { 5 }$ 
H) $\frac { 1 } { 10 }$

Quizplus · Accepted Answer

The Maclaurin series for $\cos(x^2)$ is $1 - \frac{x^4}{2!} + \frac{x^8}{4!} - \cdots$. Integrating term by term to find the series for $f(x)$, we get $x - \frac{x^5}{5\cdot2!} + \frac{x^9}{9\cdot4!} - \cdots$. The coefficient of $x^5$ is $-\frac{1}{10}$.

Find the Coefficient Of x5x ^ { 5 }x5 In the Maclaurin Series For

Find the Coefficient Of $x ^ { 5 }$ In the Maclaurin Series For