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ana111790
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Homework Statement
[tex]\int\sqrt{49-4x^2}[/tex]
I have done the integration to the best of my ability. My teacher requires that I differentiate the answer to get back to the original integrand. Can someone please check my work and help me along with the differentiation?
Homework Equations
U substitutions, and double angle formula ((sinx)^2=1/2 - (cosx)/2
The Attempt at a Solution
[tex]7\int\sqrt{1-(2x/7)^2}[/tex]
cosu=2x/7
-sinu du=2/7 dx
u=arccos(2x/7)
Integral Rewritten as:
[tex]\int[/tex][tex]7\sqrt{1-cos^2}* -7(sinu)/2 du[/tex]
[tex]7\sqrt{1-cos^2}= sinu[/tex] therefore:
=[tex]-49/2\int(sin^2u)du[/tex]
=[tex]-49/2\int(1-cos^2)/2 du[/tex]
=[tex]-49x/4 + 49(sinu)/2 + C[/tex]
=[tex]-49x/4 + 49(sin(arccos(2x/7)))[/tex])/4 + C[/tex]
Differentiation:
[tex]-49/4 + [49(cos(arccos(2x/7))]/(4 \sqrt{1-(2x/7)^2}) * 4/49[/tex]
=[tex]-49/4 + (2x)/(7\sqrt{49-4x^2})[/tex]Does this make sense at all?
How can I get to the original integrand given [tex]\int\sqrt{49-4x^2}[/tex]
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