# An integral by parts quesiton

• vande060
In summary, the conversation is about solving the integral S cos(x^1/2)dx using substitution and integration by parts. The suggested substitution is u = x^1/2 and the resulting integral is 2 S cos(u)/u du. However, the correct substitution is t = √x and the resulting integral is 2 ∫ t cos(t) dt. This can be easily solved using integration by parts.

## Homework Statement

I have to solve this integral

S cos(x^1/2)dx

where S is the integral symbol

## The Attempt at a Solution

the book tells me to use substitution and then integrate by parts

so i say u = x^1/2
du = 1/2*x^-1/2

then i can write 2 S (cos(u)du)/ x^1/2

where S in the integral sign

from here i think i can substitute the x^1/2 in the denominator by u because of the definition u = x^1/2

after the last substitution my integral would look like 2 S cos(u)/u

is this even close to right

vande060 said:

## Homework Statement

I have to solve this integral

S cos(x^1/2)dx

where S is the integral symbol

## The Attempt at a Solution

the book tells me to use substitution and then integrate by parts

so i say u = x^1/2
du = 1/2*x^-1/2 dx

then i can write 2 S (cos(u)du)/ x^1/2

where S in the integral sign

from here i think i can substitute the x^1/2 in the denominator by u because of the definition u = x^1/2

after the last substitution my integral would look like 2 S cos(u)/u du

is this even close to right

There. Fixed the missing part. You'll never get an elementary function instead of the question mark below

$$\int\frac{\cos u}{u} {}du= ? + C$$

However, the computation you made is wrong. <u> should be in the numerator, so the <exotic> part won't apply.

Last edited:
$$t=\sqrt{x} \implies t^2=x \implies 2tdt=dx$$

$$2 \int \, t \, cos(t) \, dt$$

A quick application of integration by parts will kill it.

AfterSunShine said:
$$t=\sqrt{x} \implies t^2=x \implies 2tdt=dx$$

$$2 \int \, t \, cos(t) \, dt$$

A quick application of integration by parts will kill it.

oh wow i didnt even think of that, good suggestion and thank you. i can finish the integration by parts no problem

## 1. What is an integral by parts question?

An integral by parts question is a type of calculus problem that involves finding the integral of a product of two functions.

## 2. How do you solve an integral by parts question?

To solve an integral by parts question, you must use the integration by parts formula, which states that the integral of a product of two functions is equal to the first function times the integral of the second function, minus the integral of the derivative of the first function times the integral of the second function.

## 3. What is the purpose of using integration by parts?

The purpose of using integration by parts is to simplify the integration of products of functions that cannot be integrated using other methods, such as substitution or the power rule.

## 4. Are there any tricks or tips for solving an integral by parts question?

Yes, one tip is to choose the first function to be the one that becomes simpler after taking the derivative, and the second function to be the one that becomes more complicated after integration.

## 5. Can an integral by parts question have multiple steps?

Yes, an integral by parts question can have multiple steps, especially if the resulting integral after applying the formula is still not easily solvable. In such cases, you can continue applying the formula until the integral becomes manageable.