Evaluate : ∫ (x+cos6x)/(3x2 + sin6x)dx\(\int\frac{x+cos\,6x}{3x^2+s

1.	Evaluate : ∫ (x+cos6x)/(3x2 + sin6x)dx\(\int\frac{x+cos\,6x}{3x^2+sin\,6x}\)dx
Answer» Given, ∫ (x+cos6x)/(3x² + sin6x)dx Let 3x² + sin6x = t ⇒ \(\frac{d}{dx}\) (3x² + sin6x) = dt ⇒ 6x + cos 6x. 6 = dt ⇒ x + cos 6x = \(\frac{dt}{6}\) Substituting the values, = \(\int\frac{1}{6t}\) dt = \(\frac{1}{6}\)log t +c = \(\frac{1}{6}\)log(3x² + sin6x) +c

Evaluate : ∫ (x+cos6x)/(3x2 + sin6x)dx\(\int\frac{x+cos\,6x}{3x^2+sin\,6x}\)dx

Answer»

Given,

∫ (x+cos6x)/(3x² + sin6x)dx

Let 3x² + sin6x = t

⇒ \(\frac{d}{dx}\) (3x² + sin6x) = dt

⇒ 6x + cos 6x. 6 = dt

⇒ x + cos 6x = \(\frac{dt}{6}\)

Substituting the values,

= \(\int\frac{1}{6t}\) dt

= \(\frac{1}{6}\)log t +c

= \(\frac{1}{6}\)log(3x² + sin6x) +c