Solving Definite Integral: x/sqrt(3x^2 + 4)

In summary, The conversation is about finding the integral of x over the square root of 3x^2 + 4 using u-substitution. The approach is to let u = 3x^2 + 4 and substitute it into the integral, adjusting the limits appropriately and changing dx to du. The final answer is 1/6 times the integral of 1 over the square root of u.
  • #1
cogs24
30
0
hi guys
yeh, I am still going through revision, and I am also stuck on this question.

*integral sign*(upper limit 4, lower limit 2) x/sqrt(3x^2 + 4).dx

When i look at this, i think of letting u = 3x^2 + 4, then bringing that sqrt to the top,solving for dx, and then find the integral, and sub the values into find the definite answer
Could someone clarify this for me
Thanx
 
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  • #2
a U substitution would take care of it, but you would still be left with [itex] u^{-\frac{1}{2}}[/tex], so you wouldn't be bringing the sqrt to the top unless you rationalized the denominator.

[tex] \int \frac{x}{\sqrt{3x^2+4}}\ dx = \frac{1}{6} \int \frac{1}{\sqrt{u}} \ du \ where \ u = 3x^2+4 [/tex]

Dont forget to adjust your integration limits.
 
  • #3
Your approach is basically right. You want

[tex]\int^4_2\frac{ x}{\sqrt{3x^2 + 4}} dx[/tex]

Put

[tex]u= 3x^2 + 4[/tex]

and sub it in. Don't forget about changing the dx to a du in the integral.
 

1. What is the formula for solving a definite integral?

The formula for solving a definite integral is:
ab f(x) dx = F(b) - F(a)
where a and b are the lower and upper limits of integration, f(x) is the integrand, and F(x) is the antiderivative of f(x).

2. How do I solve a definite integral using substitution?

To solve a definite integral using substitution, follow these steps:
1. Identify a part of the integrand that can be substituted with a new variable u.
2. Determine the differential du by taking the derivative of u with respect to x.
3. Rewrite the original integral in terms of u and du.
4. Solve the new integral by finding the antiderivative of f(u) with respect to u.
5. Substitute back in the original variable x and evaluate the integral using the given limits.

3. Can I use trigonometric substitution to solve this definite integral?

Yes, you can use trigonometric substitution to solve this definite integral.
Substituting u = √(3x^2 + 4) with x = √(4/3)sinθ will allow you to rewrite the integral in terms of trigonometric functions.
After finding the antiderivative, you can then substitute back in x and evaluate the integral using the given limits.

4. What is the relationship between definite and indefinite integrals?

A definite integral is the numerical value of the area under a curve between two given limits.
An indefinite integral, on the other hand, is the general antiderivative of a function.
The definite integral can be found by evaluating the indefinite integral at the given limits.

5. How can I check if my solution to a definite integral is correct?

You can check if your solution to a definite integral is correct by taking the derivative of your answer and seeing if it matches the original integrand.
You can also use a graphing calculator to plot the original function and the antiderivative to visually confirm the solution.

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