# Definite integral of square root+cube root

• MHB
• tree21c
In summary, The conversation is about a user asking for help with solving an integral that they have attempted using substitution but were unsuccessful. They mention that Wolframalpha gave a result of 6, but they are unsure of how to proceed. Another user suggests using a formula for solving integrals involving square roots. The original user then asks if the integral can be solved using elementary functions, to which the second user responds that they do not believe it can.
tree21c
Dear all,

View attachment 7576

I tried integral by substitution, but failed.

Wolframalpha shows the result is 6, but I don't know how to proceed it.

Can it be solved by elementary function?

#### Attachments

• integral.png
1.9 KB · Views: 77
Last edited:
Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

tree21c said:
Can it be solved by elementary function?

No; I don't think so.

Try applying the formula

$$\int\sqrt {x^2+a^2}dx=\frac{x\sqrt {x^2+a^2}}{2}+\frac{a^2}{2}ln(x+\sqrt {x^2+a^2})$$

## 1. What is the formula for finding the definite integral of square root+cube root?

The formula for finding the definite integral of square root+cube root is: $\int_{a}^{b}\sqrt{x}+x^{\frac{1}{3}}dx=\frac{2}{3}\left [ x^{\frac{3}{2}}+\frac{3}{4}x^{\frac{4}{3}}\right ]_{a}^{b}$

## 2. How do I solve a definite integral of square root+cube root?

To solve a definite integral of square root+cube root, you can use the formula for definite integrals and plug in the upper and lower limits of integration for a and b. Then, simplify the resulting expression using basic algebra to get the final answer.

## 3. What is the meaning of a definite integral of square root+cube root?

A definite integral of square root+cube root represents the area under the curve of the given function between two specific points on the x-axis. It can also be interpreted as the net change in the function over a certain interval.

## 4. Can a definite integral of square root+cube root have negative values?

Yes, a definite integral of square root+cube root can have negative values. This occurs when the function has negative values between the upper and lower limits of integration, or when the area under the curve is below the x-axis.

## 5. How is the definite integral of square root+cube root related to the antiderivative?

The definite integral of square root+cube root is related to the antiderivative by the Fundamental Theorem of Calculus, which states that the definite integral is equal to the difference of the antiderivative evaluated at the upper and lower limits of integration. In other words, the antiderivative is the reverse process of finding the definite integral.

• Calculus
Replies
3
Views
2K
• Calculus
Replies
11
Views
814
• Calculus
Replies
3
Views
2K
• Calculus
Replies
6
Views
2K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
6
Views
1K
• Calculus
Replies
1
Views
2K
• Calculus
Replies
8
Views
771
• Calculus
Replies
1
Views
1K
• Calculus
Replies
7
Views
2K