Integrating sqrt(x) cos(sqrt(x)) dx

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• Phys12
In summary, the conversation discusses the use of integration by parts to solve the integral of sqrt(x) cos(sqrt(x)) dx. One person attempted to solve it by differentiating sin(sqrt(x)), but this method was incorrect. The correct solution involves using integration by parts, which leads to the correct answer of 2x sin(sqrt(x)) + 2 cos(sqrt(x)) sqrt(x) + C. However, when attempting to solve the original question, there is some confusion and the wrong answer is obtained. The correct solution involves using integration by parts twice and solving for the integral of sin(sqrt(x)) and cos(sqrt(x))/sqrt(x).
Phys12
Question: sqrt(x) cos(sqrt(x)) dx
My try:

Let dv = cos(√x) => v = 2√xsin(√x) and u = √x => du = dx/(2√x)

Using integration by parts, we get

∫√x cos(√x) dx = 2√x√x sin(√x) - ∫(2√xsin(√x) dx)/(2√x)
= 2x sin(√x) - ∫sin(√x) dx

= 2x sin(√x) + 2 cos(√x) √x

However, the answer given in the book is: http://www.wolframalpha.com/input/?i=integrate+sqrt(x)++cos(sqrt(x))

And a solution that I found says: http://www.slader.com/textbook/9780534465544-calculus-early-transcendentals/601/61-exercises/23/

Which one of us is correct? And if I am wrong, what am I doing wrong and how may I correct it?

Phys12 said:
Question: sqrt(x) cos(sqrt(x)) dx
My try:

Let dv = cos(√x) => v = 2√xsin(√x)
This is not correct.

nrqed said:
This is not correct.
I differentiated sin(sqrt(x)) and figured it out that way, why does it not work?

Phys12 said:
I differentiated sin(sqrt(x)) and figured it out that way, why does it not work?
I realize this is what you did but why did you not differentiate also the ##\sqrt{x}## in front?

nrqed said:
I realize this is what you did but why did you not differentiate also the ##\sqrt{x}## in front?
Inside the sin()? I did do that

Phys12 said:
Inside the sin()? I did do that
No, the ##\sqrt{x}## in **front** of the sine.

nrqed said:
No, the ##\sqrt{x}## in **front** of the sine.
Well, this is what I did:

d(sin(sqrt(x)))/dx = cos(sqrt(x))/2(sqrt(x) => integral cos(sqrt(x)) = 2 sqrt(x) sin(sqrt(x))

Phys12 said:
Well, this is what I did:

d(sin(sqrt(x)))/dx = cos(sqrt(x))/2(sqrt(x) => integral cos(sqrt(x)) = 2 sqrt(x) sin(sqrt(x))
You cannot move the ##\sqrt{x}## on the other side like that (in your last step).
What you showed is that
$$\frac{ d}{dx} \sin(\sqrt{x}) = \frac{\cos(\sqrt{x})}{2 \sqrt{x}}$$

which means the following

$$\int \frac{\cos(\sqrt{x})}{2 \sqrt{x}} = \sin(\sqrt{x}) +C$$ This is all that one can conclude from your calculation. So your v is incorrect

nrqed said:
You cannot move the ##\sqrt{x}## on the other side like that (in your last step).
What you showed is that
$$\frac{ d}{dx} \sin(\sqrt{x}) = \frac{\cos(\sqrt{x})}{2 \sqrt{x}}$$

which means the following

$$\int \frac{\cos(\sqrt{x})}{2 \sqrt{x}} = \sin(\sqrt{x}) +C$$ This is all that one can conclude from your calculation. So your v is incorrect
So I used integration by parts and got the correct answer, however, now, for the original question, I got this:

sqrt(x) [2 sqrt(x) sin(sqrt(x)) + 2 cos(sqrt(x))] - int (sin(sqrt(x)) dx) - int(cos(sqrt(x)) dx/sqrt(x))
and when I solve further, I get:

int (sin(sqrt(x)) dx) = -2 cos(x) x + 2 sin(x)

and int(cos(sqrt) dx/sqrt(x)) = 2 sin(sqrt(x))

and end up getting the wrong answer. :/ What am I doing wrong?

1. What is the formula for integrating sqrt(x) cos(sqrt(x)) dx?

The formula for integrating sqrt(x) cos(sqrt(x)) dx is ∫ sqrt(x) cos(sqrt(x)) dx = 2/3 * sqrt(x) * sin(sqrt(x)) + 2/3 * cos(sqrt(x)) + C.

2. How do I solve the integral of sqrt(x) cos(sqrt(x)) dx?

To solve the integral of sqrt(x) cos(sqrt(x)) dx, you can use the substitution method by letting u = sqrt(x) and du = 1/2 * x^(-1/2) dx. This will result in the integral becoming ∫ 2 * u * cos(u) du, which can be solved using integration by parts or the product rule.

3. Can I use u-substitution to solve the integral of sqrt(x) cos(sqrt(x)) dx?

Yes, u-substitution is a valid method for solving the integral of sqrt(x) cos(sqrt(x)) dx. As mentioned in the previous answer, letting u = sqrt(x) and du = 1/2 * x^(-1/2) dx will help simplify the integral and make it easier to solve.

4. Are there any other methods for solving the integral of sqrt(x) cos(sqrt(x)) dx?

Yes, there are other methods for solving the integral of sqrt(x) cos(sqrt(x)) dx such as using trigonometric identities, partial fraction decomposition, or converting the integral to a definite integral and using numerical methods for approximation.

5. Is there a visual representation of the integral of sqrt(x) cos(sqrt(x)) dx?

Yes, there are various graphing calculators or online graphing tools that can plot the function sqrt(x) cos(sqrt(x)) and show the area under the curve, which represents the integral of the function. This can help visualize the concept of integration for this particular function.

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