MHT CET · Maths · Indefinite Integration
\(\int\left[\frac{(1+\log x)}{\cos ^{2}(x \log x)}\right] d x=\)
- A \(\sin (x \log x)+c\)
- B \(\sin ^{2}(x \log x)+c\)
- C \(\log (x \log x)+c\)
- D \(\tan (x \log x)+c\)
Answer & Solution
Correct Answer
(D) \(\tan (x \log x)+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{(1+\log x)}{\cos ^{2}(x \log x)} d x\)
Put \(x \log x=t \Rightarrow\left[x \cdot \frac{1}{x}+\log x(1)\right] d x=d t \Rightarrow\) \((1-\log x) d x=d t\)
\(1=\int \frac{1}{\cos ^{2} t} d t=\int \sec ^{2} t d t=\tan t+c\)
\(=\tan (x \log x)-c\)
Put \(x \log x=t \Rightarrow\left[x \cdot \frac{1}{x}+\log x(1)\right] d x=d t \Rightarrow\) \((1-\log x) d x=d t\)
\(1=\int \frac{1}{\cos ^{2} t} d t=\int \sec ^{2} t d t=\tan t+c\)
\(=\tan (x \log x)-c\)
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