Indefinite Integrals

Step 1: Find the general form of by integrating with respect to . where is the constant of integration.
Step 2: Use the given initial condition, , to find the value of . Step 3: Substitute the value of back into the expression for .

Step 1: Identify the integral's form. It matches the special property .
Step 2: Let . Find its derivative, . Step 3: Confirm that the integrand matches the form . The form is correct. Therefore, the integral is .
Step 4: Substitute back to get the final answer.

Step 1: Use integration by parts, where . Let and .
Step 2: Differentiate and integrate . We get and .
Step 3: Substitute these into the integration by parts formula: The given statement has a positive sign before the , so it is false.

Step 1: Expand the integrand using the binomial theorem . Step 2: Integrate the expanded expression term by term using the power rule. Step 3: Simplify the final expression.