Question

If ∫(cot x – cosec² x)e^x dx = e^x f(x) + c then f(x) will be ______.

Options

• cot x + cosec x

• cot² x

• cot x

• cosec x

Answer 1

If ∫(cot x – cosec²x)e^xdx = e^x f(x) + c then f(x) will be cot x.

Explanation:

∫(cot x – cosec²x)e^x dx = e^x f(x) + c

Then, ∫(cot x – cosec²x)e^x dx

=  ∫e^x cot x dx – ∫e^x cosec² x dx

On integrating by parts

= `cot x int e^x dx - int d/(dx) cot x int e^x dx - int e^x "cosec"^2 dx + c`

= e^x cot x + ∫e^x cosec²x dx – ∫e^x cosec² dx + c

= e^x cot x + c

Hence, f(x) = cot x.