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Let F(x) = integral from 0 to x sin(3t^2) dt. Find the MacLaurin polynomial of degree 7 for F(x)

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Answer:


\displaystyle \int^x_0\sin(3t^2)\,dt\approx x^3-(27)/(42)x^7

Explanation:

Recall the MacLaurin series for sin(x)


\displaystyle \sin(x)=x-(x^3)/(3!)+(x^5)/(5!)-...

Substitute 3t²


\displaystyle \displaystyle \sin(3t^2)=3t^2-((3t^2)^3)/(3!)+((3t^2)^5)/(5!)-...=3t^2-(3^3t^6)/(3!)+(3^5t^(10))/(5!)-...

Use FTC Part 1 to find degree 7 for F(x)


\displaystyle \int^x_0\sin(3t^2)\,dt\approx(3x^3)/(3)-(3^3x^7)/(7\cdot3!)\\\\\int^x_0\sin(3t^2)\,dt\approx x^3-(27)/(42)x^7

Hopefully you remember to integrate each term and see how you get the solution!

User HunkSmile
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