scrtajntman said:Homework Statement
The problem states: Use substitution to solve:
y'=1/(x+y)^2-1Homework Equations
The Attempt at a Solution
I used the substitution of v=x+y
resulting in the answer y=[3(x-C)]^(1/3) - x
but I'm not too sure that's right
Can some help with the answer and the steps for getting it. It's just for a practice test so I'm not graded on it.
Integration by substitution is a technique used to solve integrals by replacing the original variable with a new variable that simplifies the integrand. This method is especially useful for integrals involving trigonometric functions or expressions with complex exponents.
Integration by substitution is most commonly used when the integrand contains a function and its derivative, or when the integrand can be expressed in terms of a simpler function. This method is also helpful when the integrand contains a product or quotient of functions.
The substitution variable is typically chosen to simplify the integrand. It should be a function that is easy to integrate and reduces the complexity of the expression. A good choice for the substitution variable is often the function that is inside a composite function, or a term in the integrand that is raised to a power.
The steps for integration by substitution are as follows:
1. Identify the substitution variable and its derivative in the integral.
2. Substitute the new variable and its derivative into the integral.
3. Simplify the integrand and rewrite the integral in terms of the new variable.
4. Integrate the new expression, keeping in mind to include the substitution variable in the final answer.
5. Substitute the original variable back into the solution to obtain the final answer.
Yes, there are a few common mistakes to avoid when using integration by substitution. One is forgetting to include the derivative of the substitution variable in the integral. Another is using the wrong substitution variable, which can result in a more complicated integral. It is important to double check the substitution and its derivative to ensure the correct variable is being used. Lastly, forgetting to substitute the original variable back into the final answer can also result in an incorrect solution.