009C Sample Final 2, Problem 6

(a) Express the indefinite integral  $\int \sin(x^{2})~dx$  as a power series.

(b) Express the definite integral  $\int _{0}^{1}\sin(x^{2})~dx$  as a number series.

Foundations:
What is the power series of  $\sin x?$
The power series of  $\sin x$  is   $\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}.$

Solution:

(a)

Step 1:
The power series of  $\sin x$  is   $\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}.$
So, the power series of  $\sin(x^{2})$   is
${\begin{array}{rcl}\displaystyle {}&=&\displaystyle {}\\&&\\&=&\displaystyle {}\\&&\\&=&\displaystyle {}\end{array}}$

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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