# Integration by substitution and by parts

• barksdalemc
In summary, integration by substitution involves replacing a variable in an integral with a new variable, while integration by parts involves splitting an integral into two parts and applying a specific formula. Substitution is typically used when the integrand is composed of a single function, while integration by parts is used when the integrand is composed of two functions. You can use integration by substitution when the integrand consists of a single function, and the derivative of this function is also present in the integrand. The steps for integration by substitution are identifying the inner function and its derivative, replacing them with a new variable, solving the integral, and substituting the original variable back in the final answer. Integration by parts is useful when the integrand consists of two functions,
barksdalemc
I did a few problems in integration by parts. There are two that I just can't seem to get. I've tried every type of subsitution or part I can think of.

1. e^sqrt(x)

2. sin (ln x)

Rewrite:
1. $$e^{\sqrt{x}}=\frac{2\sqrt{x}}{2\sqrt{x}}e^{\sqrt{x}}$$
2. $$\sin(\ln(x))=\frac{x}{x}\sin(\ln(x))$$
On 2., you'll get an integration "cycle"

Or let $$u = e^{\sqrt{x}}$$ and $$dv = dx$$

## 1. What is the difference between integration by substitution and by parts?

Integration by substitution involves replacing a variable in an integral with a new variable, while integration by parts involves splitting an integral into two parts and applying a specific formula. Substitution is typically used when the integrand (the function inside the integral) is composed of a single function, while integration by parts is used when the integrand is composed of two functions.

## 2. How do you know when to use integration by substitution?

You can use integration by substitution when the integrand consists of a single function, and the derivative of this function is also present in the integrand. This is known as the "u-substitution" method, where the new variable u is substituted in place of the original variable in the integral.

## 3. What are the steps for integration by substitution?

The steps for integration by substitution are as follows:

1. Identify the inner function u and its derivative in the integrand.
2. Replace the inner function and its derivative with the new variable u in the integral.
3. Substitute back in the original variable in terms of u.
4. Solve the integral using basic integration techniques.
5. Substitute the original variable back into the final answer.

## 4. When is integration by parts useful?

Integration by parts is useful when the integrand consists of two functions, and one of those functions has a known antiderivative. This method can also be used when the integrand involves products of functions, as well as when the integrand involves trigonometric functions.

## 5. What is the formula for integration by parts?

The formula for integration by parts is:

∫ u(x)v'(x) dx = u(x)v(x) - ∫ v(x)u'(x) dx

Where u(x) is the first function, v'(x) is the derivative of the second function, v(x) is the antiderivative of v'(x), and u'(x) is the derivative of u(x).

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