solution is too long and I couldn't post photos. but for integration of (tan.x)^1/2 put tanx = t^2 , later it will come out to the form 2t^2/1+t^2 = t^2+1/1+t^2 + t^2-1/1+t^2 , now integrate both separately by substituting t+ 1/t and t-1/t

Integration of √(sinx/cosx)

To integrate the expression √(sinx/cosx), we can use a technique called u-substitution. This involves substituting a new variable, u, for a part of the expression to simplify the integral. Here's how we can do it:

Step 1: Identify the substitution

Let's let u = sinx. Differentiating both sides with respect to x gives du/dx = cosx. We can solve this equation for dx to obtain dx = du/cosx.

Step 2: Rewrite the expression

Using the substitution u = sinx, we can rewrite the expression as √(u/cosx).

Step 3: Substitute and simplify

Substituting u = sinx and dx = du/cosx into the integral, we get:

∫ √(sinx/cosx) dx = ∫ √(u/cosx) (du/cosx) = ∫ √(u) du

Step 4: Integrate

Now, we need to integrate √(u) with respect to u. This integral can be evaluated by using the power rule for integration:

∫ √(u) du = (2/3)u^(3/2) + C,

where C is the constant of integration.

Step 5: Substitute back

Finally, we substitute u = sinx back into the result:

∫ √(sinx/cosx) dx = (2/3)sin^(3/2)x + C.

This is the final result of integrating √(sinx/cosx).

Summary:
To integrate √(sinx/cosx), we used the u-substitution technique. By letting u = sinx and substituting it into the expression, we simplified the integral to ∫ √(u) du. After integrating √(u) with respect to u, we substituted u = sinx back into the result to obtain the final answer: (2/3)sin^(3/2)x + C.