Question

Give an example of an integration problem that you can solve by two different methods. One of the methods μst be integration by parts. Solve the problem using the two methods.

Answer 1

An integral of x times e^x can be solved using two methods: integration by parts and recognition of derivatives. Both methods yield the same solution: x e^x - e^x + C, which demonstrates the flexibility in approaches for solving integration problems.

Step-by-step explanation:

A common example of an integration problem that can be solved by two different methods, one of which is integration by parts, is the integral of xex. We can solve this problem using integration by parts or by recognizing it as a product of two functions whose derivatives and antiderivatives are easily found.

Method 1: Integration by Parts

To apply integration by parts, we let u = x and dv = exdx. Then we find du = dx and v = ex. The formula for integration by parts is ∫udv = uv - ∫vdu. Using this formula, we obtain:

∫x exdx = x ex - ∫exdx = x ex - ex + C.

Method 2: Recognition of Derivatives

We notice that the derivative of x is 1 and the antiderivative of ex is ex. The integral of xex is simply the antiderivative of x multiplied by the derivative of ex, minus the integral of the antiderivative of ex times the derivative of x, which again gives us:

∫x exdx = x ex - ex + C.