Question

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

Answer 1

`\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!