Calculate the integrals if they converge.
10.) Integral from 1 to infinity of
X/4+X^2 dx
14.) integral from Pi/2 to Pi/4 of
Sin X / sqrt cos x dx
22.) integral from 0 to 1 of
ln x/x dx
I'm having problems with working these out to figure out if they converge or not. Any help or tips would be appreciated
To determine if these integrals converge or not, we need to apply some known tests for convergence. Here's how we can approach each integral:
10.) Integral from 1 to infinity of (x/4 + x^2) dx
To check for convergence, we can use the comparison test. We compare the given integral with a known convergent or divergent integral.
Let's consider the term x^2. As x approaches infinity, x^2 grows faster than x. So, we can ignore the term x/4 and compare the integral with ∫x^2 dx.
∫x^2 dx is a well-known convergent integral, so we can conclude that our integral converges as well.
14.) Integral from π/2 to π/4 of sin(x) / √cos(x) dx
To determine convergence here, we can use the limit comparison test. We'll compare the given integral with a known convergent integral, ∫1/√x dx, which converges.
Let's substitute y = cos(x) in the integral to simplify it. Then, we have:
∫sin(x) / √cos(x) dx = ∫(√(1 - y^2) / y) dy
Now, we shift our limits to their equivalent values in terms of y:
∫(√(1 - y^2) / y) dy from 0 to 1
To apply the limit comparison test, we need to compare the integrands, so we calculate:
lim(as y approaches 0) [√(1 - y^2) / y]
By evaluating this limit, we find that it is finite (equal to 1). Since the limit is finite and our comparison integral is convergent, we can conclude that the given integral also converges.
22.) Integral from 0 to 1 of ln(x) / x dx
To check if this integral converges, we can consider the integral as x approaches 0 and as x approaches 1 and determine if it behaves well in those limits.
As x approaches 0, ln(x) approaches negative infinity, so the integrand has a singularity there. This indicates potential divergence.
To confirm whether the integral converges or not, we consider the behavior of the integrand near x = 1. As x approaches 1, ln(x) approaches 0, and the function is well-behaved.
Therefore, the integral from 0 to 1 of ln(x) / x dx is convergent.
Remember, these are just approaches to determine convergence. It's always a good idea to double-check using other tests and methods if needed.