Which of the following integrals cannot be evaluated using a simple substitution?
the integral of the square root of the quantity x minus 1, dx
the integral of the quotient of 1 and the square root of the quantity 1 minus x squared, dx
the integral of the quotient of 1 and the square root of the quantity 1 minus x squared, dx
the integral of x times the square root of the quantity x squared minus 1, dx
#2 and #3 are the same.
#1: u = x-1
#4: u = x^2-1
To determine which of the integrals cannot be evaluated using a simple substitution, we need to analyze each integral separately.
1. The integral of the square root of the quantity (x - 1), dx:
To evaluate this integral, we can make the substitution u = x - 1. By substituting, the integral becomes ∫√u du. This can be easily integrated using the power rule for integration. Therefore, this integral can be evaluated using a simple substitution.
2. The integral of the quotient of 1 and the square root of the quantity (1 - x^2), dx:
To evaluate this integral, we can make the substitution u = 1 - x^2. By substituting, the integral becomes ∫(1/√u) du. This integral can also be integrated using a simple substitution.
3. The integral of the quotient of 1 and the square root of the quantity (1 - x^2), dx:
This is the same as the second integral mentioned above. So, this integral can also be evaluated using a simple substitution.
4. The integral of (x * √(x^2 - 1)), dx:
To evaluate this integral, we can make the substitution u = x^2 - 1. However, upon taking the derivative of u with respect to x, we would end up with du/dx = 2x. We notice that we have an extra 'x' factor in the integrand, which prevents us from performing a simple substitution. Therefore, this integral cannot be evaluated using a simple substitution.
Hence, the integral that cannot be evaluated using a simple substitution is the last one: ∫(x * √(x^2 - 1)), dx.