Panphobia said:For the function f(x) = sinx*cosx the integral of it by u-substitution could be -(cosx)^2/2 or (sinx)^2/2, which one is right for an assignment or would I need to state both? Also for a person taking first year physics, what kinds of integration will I need to know past, integration by parts, and integration by substitution?
To use u-substitution for this integral, you need to identify a part of the function that can be represented as u. In this case, let u = sinx. Then, du = cosx dx, which can replace cosx in the original function. The integral becomes ∫ u du, which is a simple power rule integration. The final answer is 1/2 * u^2 + C, where C is the constant of integration.
U-substitution is useful because it allows us to simplify complicated integrals by substituting a new variable for part of the function. This often makes the integral easier to solve using basic integration rules, such as the power rule or the substitution rule.
No, u-substitution can only be used for integrals where there is a clear substitution for u. This means that the integral must have a part of the function that can be represented as u, and the derivative of that u term must also appear in the integral.
To know when to use u-substitution, look for integrals that have a part of the function that can be represented as u. This often includes trigonometric functions, exponential functions, or polynomial functions raised to a power. You can also try to find a u term by using algebraic manipulation or by looking at the derivative of the function.
Yes, there are many other methods for solving integrals, such as integration by parts, trigonometric substitution, and partial fractions. The best method to use depends on the specific integral and the available tools and techniques. It is important to practice and be familiar with these different methods in order to choose the most efficient one for each integral.