Question

integration
e^x(1 – cotx + cosec²x)dx​

Answer 1

EXPLANATION.

⇒ ∫eˣ(1 - cot(x) + cosec²x)dx.

As we know that,

⇒ ∫eˣ[f(x) + f'(x)]dx = eˣf(x) + c.

Differentiate (1 - cot(x)) w.r.t x, we get.

⇒ - (-cosec²x)dx = dt.

⇒ cosec²x dx = dt.

⇒ ∫eˣ(1 - cot(x) + cosec²x)dx. = eˣ(1 - cot(x)) + c.

                                                                                                                     

MORE INFORMATION.

Standard integrals.

(1) = ∫0.dx = c.

(2) = ∫1.dx = x + c.

(3) = ∫k dx = kx + c, ( k ∈ R).

(4) = ∫xⁿdx = xⁿ⁺¹/n + 1 + c, (n ≠ -1).

(5) = ∫dx/x = ㏒(x) + c.

(6) = ∫eˣdx = eˣ + c.

(7) = ∫aˣdx = aˣ/㏒(a) + c = aˣ㏒(e) + c.