Question

Evaluate: $\int \frac{dx}{x\cos x}$

• A. $\ln x+\frac{x^2}{2}+\frac{3x^3}{3}+\cdots$
• B. $\ln x-\frac{x^2}{4}+\frac{4x^4}{16}-\cdots$
• C. $\ln x+\frac{x^2}{4}+\frac{5x^4}{96}+\cdots$
• D. $\ln x-\frac{x^2}{3}+\frac{x^4}{9}+-\cdots$

Answer 1

The correct option is C $\ln x+\frac{x^2}{4}+\frac{5x^4}{96}+\cdots$

I = $\int \frac{dx}{x\cos x}$

=$\int \frac{secx}{x}dx$

Maclaurin series for

$secx=1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...$

=$\int \frac{1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...}{x}dx$

=$\ln x+\frac{x^2}{4}+\frac{5x^4}{96}+\cdots$