009C Sample Final 2, Problem 6 Detailed Solution

(a) Express the indefinite integral  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \sin(x^2)~dx}   as a power series.

(b) Express the definite integral  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^1 \sin(x^2)~dx}   as a number series.

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Background Information:
What is the power series of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x?}
The power series of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin x}   is   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}.}

Solution:

(a)

Step 1:
The power series of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin x}   is   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}.}
So, the power series of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin(x^{2})}   is
${\begin{array}{rcl}\displaystyle {\sin(x^{2})}&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}(x^{2})^{2n+1}}{(2n+1)!}}}\\&&\\&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+2}}{(2n+1)!}}.}\end{array}}$

Step 2:
Now, to express the indefinite integral as a power series, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\int \sin(x^{2})~dx}&=&\displaystyle {\int \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+2}}{(2n+1)!}}~dx}\\&&\\&=&\displaystyle {\sum _{n=0}^{\infty }\int {\frac {(-1)^{n}x^{4n+2}}{(2n+1)!}}~dx}\\&&\\&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+3}}{(4n+3)(2n+1)!}}.}\end{array}}}

(b)

Step 1:
From part (a), we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \sin(x^{2})~dx=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+3}}{(4n+3)(2n+1)!}}.}
Now, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{0}^{1}\sin(x^{2})~dx=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+3}}{(4n+3)(2n+1)!}}{\bigg |}_{0}^{1}.}

Step 2:
Hence, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\int _{0}^{1}\sin(x^{2})~dx}&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}(1)^{4n+3}}{(4n+3)(2n+1)!}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}(0)^{4n+3}}{(4n+3)(2n+1)!}}}\\&&\\&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(4n+3)(2n+1)!}}-0}\\&&\\&=&\displaystyle {\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(4n+3)(2n+1)!}}.}\end{array}}}

Final Answer:
(a)     Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{4n+3}}{(4n+3)(2n+1)!}}}
(b)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty \frac{(-1)^n}{(4n+3)(2n+1)!}}

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