∫ $\frac{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{x}-\mathit{a}\right)}{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{x}+\mathit{a}\right)}$ dx
Let (x + a) = t ⇒ dx = dt
∴ I = ∫ $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{t}-2\mathit{a}}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}$ dt
= ∫ $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathit{c}\mathit{o}\mathit{s}2\mathit{a}-\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathit{s}\mathit{i}\mathit{n}2\mathit{a}}{\mathit{s}\mathit{i}\mathit{n}\mathit{t}}$ dt
= ∫ cos 2a - cot t sin 2a dt
= cos 2a t - sin 2a log |sin t| + C
= cos 2a x + a - sin 2a log |sin (x + a)| + C
OR
∫ $\frac{5\mathit{x}-2}{1+2\mathit{x}+3{\mathit{x}}^2}$ dx
= 5 ∫ $\frac{\mathit{x}-\frac{2}{5}}{1+2\mathit{x}+3{\mathit{x}}^2}$ dx
= $\frac{5}{6}$ ∫ $\frac{6\mathit{x}-\frac{12}{5}}{1+2\mathit{x}+3{\mathit{x}}^2}$ dx
= $\frac{5}{6}$ ∫ $\frac{6\mathit{x}+2-\frac{12}{5}-2}{1+2\mathit{x}+3{\mathit{x}}^2}$ dx
= $\frac{5}{6}$ ∫ $\frac{6\mathit{x}+2-\frac{22}{5}}{1+2\mathit{x}+3{\mathit{x}}^2}$ dx
= $\frac{5}{6}$ ∫ $\frac{6\mathit{x}+2}{1+2\mathit{x}+3{\mathit{x}}^2}$ - $\frac{5}{6}\times \frac{22}{5}$ ∫ $\frac{1}{3\left[{\left(\mathit{x}+\frac{1}{3}\right)}^2+\frac{2}{9}\right]}$ dx
= $\frac{5}{6}$ log |1 + 2x + $3{\mathit{x}}^2$| - $\frac{11}{9}$ ∫ $\frac{1}{{\left(\mathit{x}+\frac{1}{3}\right)}^2+\frac{2}{9}}$ dx
= $\frac{5}{6}$ log |1 + 2x + $3{\mathit{x}}^2$| - $\frac{11}{9}\times \frac{3}{\sqrt{2}}$ $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\frac{\left(\mathit{x}+\frac{1}{3}\right)}{\frac{\sqrt{2}}{3}}$ + C
= $\frac{5}{6}$ log |1 + 2x + $3{\mathit{x}}^2$| - $\frac{11}{3\sqrt{2}}$ × $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{3\mathit{x}+1}{\sqrt{2}}\right)$ + C