(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.
| 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 (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}
is 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)^nx^{2n+1}}{(2n+1)!}.}
|
Solution:
(a)
| Step 1:
|
| 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}
is 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)^nx^{2n+1}}{(2n+1)!}.}
|
| So, 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^2)}
is
|
|
|
| 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 \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)^nx^{4n+2}}{(2n+1)!}.} \end{array}}
|
|
|
| Step 2:
|
| Now, to express the indefinite integral as a power series, we have
|
|
|
| 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 \begin{array}{rcl} \displaystyle{\int \sin(x^2)~dx} & = & \displaystyle{\int \sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}~dx}\\ &&\\ & = & \displaystyle{\sum_{n=0}^\infty \int \frac{(-1)^nx^{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 (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=\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}.}
|
| Now, we have
|
|
| Step 2:
|
| Hence, we have
|
|
|
| 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 \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 (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 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)!}}
|
Return to Sample Exam