What did you use for the substitution? What did you use for the parts of the integration by parts?Shawn Garsed said:Homework Statement
Evaulate the integral of (4-x2)-3/2.
Homework Equations
integration by parts
u-replacement
The Attempt at a Solution
Too long to write it all down, but I used the aforementioned equations. Basically, I'd like somebody to give me a hint about how or where to start, since everything I tried so far hasn't worked.
To integrate a reciprocal function, you need to use the power rule for integration. This means that you need to raise the function to the power of -1 and then add 1 to the power. For example, if your function is f(x) = 1/x, then you would raise it to the power of -1 and add 1, resulting in f(x) = x^-1 + 1. Then you can apply the power rule, which states that the integral of x^n is (x^n+1)/n+1. In this case, the integral of x^-1 would be (x^-1+1)/-1+1, which simplifies to -x^-1 + C.
The inverse function of a reciprocal function is simply the reciprocal function itself. This means that if your original function is f(x) = 1/x, then the inverse function would be f^-1(x) = x. This is because when you plug in x into the inverse function, you would get 1/x, which is the original function.
Yes, substitution can be used to integrate a reciprocal function. This method involves substituting a variable for the function inside the integral, and then using the chain rule to solve for the integral. For example, if your function is f(x) = 1/x, you could substitute u = 1/x, which would result in du = -1/x^2 dx. Then you can solve for the integral using the new variable u.
The limits of integration for a reciprocal function depend on the specific problem or context in which the function is being used. In general, the limits of integration would be the values of x for which the function is defined. For example, if your function is f(x) = 1/x, the limits of integration could be any values of x where x ≠ 0.
Integration of a reciprocal function has many real-life applications, such as in physics and engineering. One example is in the calculation of work done by a variable force. The work done is equal to the integral of the reciprocal of the force with respect to the displacement. Reciprocal functions are also commonly used in economic and financial analysis, such as in the calculation of marginal utility and elasticity of demand.