# Integration by Parts: Solving Homework

• crm08
In summary, the problem is asking to find the integral of arcsin(x) and the formula to use is \int v du=uv-\int udv. After some confusion with selection of u and v, the correct solution is x*arcsin(x) + sqrt(1-x^2) + C.
crm08

## Homework Statement

$$\int(sin(x)^{-1}), dx$$

## Homework Equations

*By Parts Formula: f(x)g(x) - $$\int(g(x) f'(x)) dx$$

Also for d/dx sin(x)^{-1} I used 1/sqrt(1-x^{2})

## The Attempt at a Solution

Just started learning this method, I tried letting f(x) = sin(x)^{-1} and g(x) = dx but nothing really simplified, can someone help with selecting the correct g(x) and f(x), Thanks

Last edited:
crm08 said:

## Homework Statement

$$\int(sin(x)^{-1}), dx$$

Is the function you're trying to integrate here the inverse function of sin(x)? If so, it should be denoted $$\arcsin(x)=\sin^{-1}(x)$$

## Homework Equations

*By Parts Formula: f(x)g(x) - $$\int(g(x) f'(x)) dx$$

This isn't a formula, since you haven't specified what it is equal to! The formula I would use is $$\int v du=uv-\int udv$$. Is this the formula you have been taught? If not, what is the formula you have been taught?

Yes, the problem is asking for the integral of arcsin(x), and also yes, that is the formula we are using, my "u's" and "v's" look a lot alike sometimes so I replaced them with f(x) and g(x), sorry about the confusion

crm08 said:
Yes, the problem is asking for the integral of arcsin(x), and also yes, that is the formula we are using, my "u's" and "v's" look a lot alike sometimes so I replaced them with f(x) and g(x), sorry about the confusion

Ok, so your selection was $u=\sin^{-1}(x) \,\, , \,\, dv=dx$, right? So, what went wrong? This is the choice that I would make!

Ok nevermind I got it now, I was working towards an answer to this problem that my 89 gave me but I typed it in wrong, I see how to to it now, the answer being:

x*arcsin(x) + sqrt(1-x^2)

Plus a constant

## What is integration by parts?

Integration by parts is a technique used in calculus to solve integrals of functions that are products of other functions. It involves breaking down the integral into smaller, simpler parts and using the product rule of differentiation to solve it.

## When should integration by parts be used?

Integration by parts should be used when the integral involves a product of two functions, where one function is easy to integrate but the other is not. It is also useful when the integral involves a function that repeats itself after being differentiated multiple times.

## What is the formula for integration by parts?

The formula for integration by parts is ∫udv = uv - ∫vdu, where u and v are functions and du and dv are their respective derivatives.

## What are the steps to solve an integral using integration by parts?

The steps to solve an integral using integration by parts are: 1) Identify u and dv in the integral, 2) Find du and v by differentiating and integrating u and dv respectively, 3) Substitute the values of u, du, v, and dv into the integration by parts formula, and 4) Simplify and solve the resulting integral.

## What are some common mistakes to avoid when using integration by parts?

Some common mistakes to avoid when using integration by parts are: 1) Choosing the wrong u and dv values, 2) Forgetting to include the minus sign in the integration by parts formula, 3) Not simplifying the resulting integral, and 4) Forgetting to add the constant of integration at the end.

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