Question

Tính tích phân bất định hàm số lượng giác : 

\(I=\int\frac{1}{a.\sin x+b.\cos x}dx\)

Answer 1

\(I=\frac{1}{\sqrt{a^2+b^2}}\int\frac{dx}{\sin\left(x+\alpha\right)}=\frac{1}{\sqrt{a^2+b^2}}\int\frac{dx}{2\sin\frac{x+\alpha}{2}.\cos\frac{x+\alpha}{2}}=\frac{1}{\sqrt{a^2+b^2}}\int\frac{dx}{2\tan\frac{x+\alpha}{2}.\cos^2\frac{x+\alpha}{2}}\)

\(\Rightarrow\frac{1}{\sqrt{a^2+b^2}}\int\frac{d\left(\tan\frac{x+\alpha}{2}\right)}{\tan\frac{x+\alpha}{2}}=\frac{1}{\sqrt{a^2+b^2}}\ln\left|\tan\frac{x+\alpha}{2}\right|+C\)

Answer 2

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