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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 55

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Answer: \frac{1}{6}\log \left | 3x^{2}+\sin 6x \right |+c

Hints: You must know about the integral rule of trigonometric functions.

Given: \int \frac{x+\cos 6x}{3x^{2}+\sin6x}dx


\int \frac{x+\cos 6x}{3x^{2}+\sin6x}dx

Let    3x^{2}+\sin 6x=t     and differentiate both sides,\left [ \frac{d}{dx}\sin x=\cos x \right ]

(6x+6\cos 6x)dx=dt

(x+\cos 6x)dx=\frac{dt}{6}

\begin{aligned} &I=\int \frac{d t}{6 t}=>\frac{1}{6} \log (t)+c \\ &=\frac{1}{6} \log \left|3 x^{2}+\sin 6 x\right|+c \quad\quad\quad\quad\quad\left[\int \frac{1}{x} d x=\log x+c\right]\\ \end{aligned}

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