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

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

We want to find the value of the definite integral:

We can integrate using a substitution:

Substituting into the original integral we get;

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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.

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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.

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