The integrand does not behave as 1/[x^(1/3)*x^(1/2)]. Even if it did, 1/x^(1/6) becomes infinitely large as x approaches zero. You're probably thinking of the behavior of a p series, where n is growing infinitely large.a1010711 said:Homework Statement
the integral from 0 to 1 given:
1 / [ (x^1/3)(x^2+2x)^1/2 ] dx
please explain,, thank you!
The Attempt at a Solution
At 0 the integrand behaves as 1/x^1/3*x^1/2 = 1/x^1/6 which is convergent as the exponent is <1
Substitute WHAT into the integral?lanedance said:could maybe do it 2 ways based on the integral 1->inf of 1/x^n converging iff n>1
2 ways potentially
substitute straight into your integral, changing integration from 0to1 to 1 to infinty, and then use some comparison theorems
Lanedance, I'm not following you here.lanedance said:or substitute into the integral 1->inf of 1/x^n which will change it to 0 to 1, should give you a condition based on n for this integral to converge
i'm thinking it probably does converge at this stage
This is a common question asked when evaluating integrals. The answer depends on the specific integral and its limits of integration. Some integrals converge, meaning they have a finite value, while others diverge, meaning they do not have a finite value.
To determine if an integral converges or diverges, you can use various methods such as the comparison test, the limit comparison test, or the integral test. These methods involve evaluating the behavior of the integral as the limits of integration approach infinity.
A convergent integral has a finite value, meaning it converges to a specific number. On the other hand, a divergent integral does not have a finite value and either approaches infinity or oscillates between different values as the limits of integration increase.
Yes, it is possible for an integral to converge for certain values of the variable and diverge for others. This is known as conditional convergence and occurs when the integral has both positive and negative terms, and the sum of these terms is finite for some values of the variable but infinite for others.
No, there is no general rule for determining the convergence or divergence of integrals. The behavior of an integral depends on various factors such as the function being integrated, the limits of integration, and the method used to evaluate the integral. It is important to use different methods to determine the convergence or divergence of integrals and to confirm the results using multiple approaches.