The given indefinite integral ∫cotxcosec2xdx
i) The substitution rule
If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here g(x) is substituted as u and then the derivative g’(x)dx =du.
iii) Indefinite integral
∫xn dx=xn+1n+1+C n≠-1
The given integral is
Here, use the substitution method because the differential of the function cotx, -cosec2x apart from the negative sign present in the integral.
Differentiating with respect to x.
Therefore, the given integral becomes
By using concept iii),
Now, resubstitute cotx=u.
Hence, the solution becomes