integration of sqrt of tan(x)

how can i wirte a story problem and show my solution in the same time

First, if you have a question, it is much better to put it in as a separate post in <Post a New Question> rather than attaching it to a previous question, where it is more likely to be overlooked.

Type it all out?

To solve the integral of √tan(x), we can use a technique called trigonometric substitution.

Here's how to approach it:

1. Start by letting tan(x) = u², which implies sec²(x) dx = du. This substitution reduces the integral to a simpler form involving u.
2. Express sec²(x) in terms of u: Using the identity sec²(x) = 1 + tan²(x), we have 1 + u² dx = du.
3. Rearrange the equation to isolate dx: dx = du / (1 + u²).
4. Substitute back into the integral: Instead of integrating with respect to x, we now integrate with respect to u. Hence, our integral becomes: ∫ √u² / (1 + u²) du.

Now, we need to simplify the integral using partial fraction decomposition:

5. Split the fraction: Rewrite the integrand as √u² / (1 + u²) = (A + Bu) / (1 + u²), where A and B are constants to be determined.
6. Find the common denominator of the resulting fraction and equate the numerators: √u² = (A + Bu).
7. Expand the equation and group like terms: u = Au + Bu².
8. Equate the coefficients of like powers of u: 1 = A and 0 = B.

With A = 1 and B = 0, the integral simplifies to:

∫ √u² / (1 + u²) du
= ∫ du
= u + C.

9. Substitute u back in terms of x: Since we initially made the substitution tan(x) = u², solving for u gives u = √tan(x).
10. Finally, we have ∫ √tan(x) dx = √tan(x) + C, where C is the constant of integration.

That's it! The integral of √tan(x) is √tan(x) + C, where C is the constant of integration.