# Evaluate # int cos(x^2) dx # using a power series?

# int \ cos(x^2) \ dx = c+x - (x^5)/(5 * 2!) + (x^9)/(9*4!) - x^13/(13*6!) + ...#

So then:

Next up is:

By signing up, you agree to our Terms of Service and Privacy Policy

To evaluate ( \int \cos(x^2) , dx ) using a power series, we can express ( \cos(x^2) ) as a power series using the Maclaurin series for ( \cos(x) ). The Maclaurin series for ( \cos(x) ) is ( \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ).

Substituting ( x^2 ) for ( x ), we have ( \cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n (x^2)^{2n}}{(2n)!} ). Simplifying, we get ( \cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(2n)!} ).

Now, we can integrate ( \cos(x^2) ) term by term to find ( \int \cos(x^2) , dx ).

( \int \cos(x^2) , dx = \int \left( \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(2n)!} \right) , dx )

( = \sum_{n=0}^{\infty} \left( \int \frac{(-1)^n x^{4n}}{(2n)!} , dx \right) )

( = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+1}}{(4n+1)(2n)!} + C )

where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the first two nonzero terms in Maclaurin's Formula and use it to approximate #f(1/3)# given #int e^(t^2) dt# from [0,x]?
- How do you find a power series representation for # x^2 / ( 1 - 2x )^2#?
- How do you find a power series representation for # f(z)=z^2 # and what is the radius of convergence?
- What is the Maclaurin series of #f(x) = cos(x)#?
- How do you find the taylor series series for #f(x) = x^4 - x^2 + 1# at c=-1?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7