The formula for integrating by parts is ∫u dv = uv - ∫v du, where u and v are functions and du and dv are their derivatives.
When using the integration by parts method, the choice of u and dv depends on the complexity of the function and its derivatives. Generally, u should be chosen as the function that becomes simpler after differentiation and dv should be chosen as the function that can be easily integrated.
The steps for integrating by parts are as follows:
Using the formula for integration by parts, we can let u = arctan(√x) and dv = dx. This gives us du = 1/(1+x)dx and v = x. Substituting these values into the formula, we get: ∫arctan(√x)dx = xarctan(√x) - ∫x/(1+x)dx We can then solve the integral on the right side using the substitution method or by applying the quotient rule.
Integration by parts is useful when the integral involves a product of functions, such as in the case of arctan(√x)dx. It can also be used when other integration methods, such as substitution or partial fractions, are not applicable. Additionally, integration by parts is helpful for integrating functions that involve logarithms, inverse trigonometric functions, or polynomials.