How do you integrate #int x/(sqrt(1+2x^2)dx# from [0,2]?

Answer 1

# int_0^2 \ x/(sqrt(1+2x^2}) \ dx = 1 #

We want to find the value of the definite integral:

# int_0^2 \ x/(sqrt(1+2x^2}) \ dx #

We can integrate using a substitution:

Let # u=1+2x^2 => (du)/dx = 4x #
So when we substitute we will have #int \ ... x\ dx = int \ 1/4 ... \ du#, and as we have changed the variable of integration we need to change the limits of integration to match:
When #{ (x=0), (x=2) :} => { (u=1), (u=9) :}#

Substituting into the original integral we get;

# int_0^2 \ x/(sqrt(1+2x^2}) \ dx = int_1^9 \ (1/4)/(sqrt(u) \ du # # " " = 1/4 \ int_1^9 \ u^(-1/2) \ du # # " " = 1/4 \ [u^(1/2)/(1/2)]_1^9 # # " " = 1/2 \ [u^(1/2)]_1^9 # # " " = 1/2 ( sqrt(9)-sqrt(1) ) # # " " = 1/2 ( 3-1 ) # # " " = 1 #
Sign up to view the whole answer

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

Sign up with email
Answer 2

To integrate the function ( \frac{x}{\sqrt{1+2x^2}} ) over the interval [0,2], you can use the substitution method. Let ( u = 1 + 2x^2 ). Then ( du = 4x dx ). Solving for ( x ), we get ( x = \frac{\sqrt{u - 1}}{\sqrt{2}} ). Now, substitute these expressions into the integral and solve. This yields:

[ \int_{0}^{2} \frac{x}{\sqrt{1+2x^2}} , dx = \frac{1}{\sqrt{2}} \int_{1}^{5} \frac{\sqrt{u-1}}{\sqrt{u}} , du ]

Using basic integral rules and properties, integrate the expression above. This should result in an expression involving elementary functions. Once integrated, evaluate the expression at the upper and lower limits of integration (5 and 1 respectively) and subtract to find the final answer.

Sign up to view the whole answer

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

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7