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| | <span class="exam">(b) Express the definite integral <math style="vertical-align: -14px">\int_0^1 \sin(x^2)~dx</math> as a number series. | | <span class="exam">(b) Express the definite integral <math style="vertical-align: -14px">\int_0^1 \sin(x^2)~dx</math> as a number series. |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009C Sample Final 2, Problem 6 Solution|'''<u>Solution</u>''']] |
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| − | |What is the power series of <math style="vertical-align: -1px">\sin x?</math>
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| − | | The power series of <math style="vertical-align: -1px"> \sin x</math> is <math>\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.</math>
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| − | '''Solution:''' | + | [[009C Sample Final 2, Problem 6 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |The power series of <math style="vertical-align: -1px"> \sin x</math> is <math>\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.</math>
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| − | |So, the power series of <math style="vertical-align: -5px">\sin(x^2)</math> is
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\sin(x^2)} & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n(x^2)^{2n+1}}{(2n+1)!}}\\
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| − | &&\\
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| − | & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}.}
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| − | \end{array}</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Now, to express the indefinite integral as a power series, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\int \sin(x^2)~dx} & = & \displaystyle{\int \sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}~dx}\\
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| − | &&\\
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| − | & = & \displaystyle{\sum_{n=0}^\infty \int \frac{(-1)^nx^{4n+2}}{(2n+1)!}~dx}\\
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| − | &&\\
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| − | & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}.}
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| − | \end{array}</math>
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| − | |}
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)''' <math>\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}</math>
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| − | | '''(b)''' <math></math>
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| − | |}
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| | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
