ʃ ( tanx + cotx )² dx
( tanx + cotx )²
= tan^2 + 2 + cot^2
= sec^2 - 1 + 2 + csc^2 - 1
Now it should be easy.
thank you
To evaluate the integral ∫(tanx + cotx)² dx, we can expand the square and then integrate each term separately.
Let's start by expanding the square:
(tanx + cotx)² = tan²x + 2tanx*cotx + cot²x
Now, let's integrate each term:
∫tan²x dx:
To integrate tan²x, we can use the trigonometric identity: tan²x = sec²x - 1.
So, we have:
∫tan²x dx = ∫(sec²x - 1) dx = ∫sec²x dx - ∫dx
The integral of sec²x is a well-known integral and is equal to tanx. The integral of dx is x. So, the integral becomes:
∫tan²x dx = tanx - x
Next, let's integrate the middle term:
∫2tanx*cotx dx:
We can simplify 2tanx*cotx to 2.
So, we have:
∫2tanx*cotx dx = ∫2 dx = 2x
Finally, let's integrate the last term:
∫cot²x dx:
To integrate cot²x, we can use the trigonometric identity: cot²x = csc²x - 1.
So, we have:
∫cot²x dx = ∫(csc²x - 1) dx = ∫csc²x dx - ∫dx
The integral of csc²x is a well-known integral and is equal to -cotx. The integral of dx is x. So, the integral becomes:
∫cot²x dx = -cotx - x
Now, let's put everything together:
∫(tanx + cotx)² dx = ∫(tan²x + 2tanx*cotx + cot²x) dx
= ∫(tanx - x + 2x - cotx - x) dx
= ∫(tanx - cotx) dx
= -cotx - x + 2x
= x - cotx
Therefore, the solution to the integral ∫(tanx + cotx)² dx is x - cotx.