Taylor series : binomial series

Using the 3 first terms of the binomial series, approximate the following indefinite integral: (8+x)^1/3
Using the first three terms of power series for the function f(x)=cos x , estimate the following integral: f(x)=cos(x^2/3) dx
thanks

1) Note that (8 + x)^(1/3)
= [8(1 + x/8)]^(1/3)
= 2(1 + x/8)^(1/3)
= 2 [1 + (1/3)(x/8) + ((1/3)(1/3 - 1)/2!) (x/8)^2 + ...], via binomial series
= 2 + (1/12) x - (1/288) x^2 + ...

So, ∫ (8 + x)^(1/3) dx
= ∫ [2 + (1/12) x - (1/288) x^2 + ...] dx
= [2x + (1/24) x^2 - (1/864) x^3 + ... ] + C.
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2) cos((1/3) x^2)
= 1 - ((1/3) x^2)^2 / 2! + ((1/3) x^2)^4 / 4! - ...
= 1 - (1/18) x^4 + (1/1944) x^8 - ...

So, ∫ cos((1/3) x^2) dx
= ∫ [1 - (1/18) x^4 + (1/1944) x^8 - ...] dx
= [x - (1/90) x^5 + (1/17496) x^9 - ...] + C.

I hope this helps!