Math, asked by Jaisgrewal5470, 2 months ago

integration of dx by sinx + √3cosx

Answers

Answered by amansharma264
4

EXPLANATION.

\implies \displaystyle \int \dfrac{dx}{sin(x) + \sqrt{3} cos(x)}

As we know that,

Multiply and divide numerator and denominator by 1/2, we get.

\implies \displaystyle \dfrac{1}{2} \int \dfrac{dx}{\bigg(\dfrac{sin(x)}{2}  + \dfrac{\sqrt{3} cos(x)}{2} \bigg)}

\implies \displaystyle \dfrac{1}{2} \int \dfrac{dx}{sin \bigg(x + \dfrac{\pi}{3} \bigg)}

\implies \displaystyle \dfrac{1}{2} \int cosec \bigg(x + \dfrac{\pi}{3} \bigg)dx

\implies \displaystyle \dfrac{1}{2}  \ log \bigg| tan \bigg( \dfrac{x}{2}  + \dfrac{\pi}{6} \bigg) \bigg| + C

\implies \displaystyle \int \dfrac{dx}{sin(x) + \sqrt{3} cos(x)} \ = \dfrac{1}{2} \ log \bigg| tan \bigg(\dfrac{x}{2}  + \dfrac{\pi}{6} \bigg) \bigg| + C.

                                                                                                                   

MORE INFORMATION.

(1) = ∫0.dx = c.

(2) = ∫1.dx = x + c.

(3) = ∫k dx = kx + c, (x ∈ 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.

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