Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Consider a real valued continuous function $f$ such that $f(x) = \sin x + \int_{-\pi/2}^{\pi/2} (\sin x + tf(t))dt$. If $M$ and $m$ are maximum and minimum value of the function $f$, then the value of $M/m$ is _____.
Step-by-Step Solution
Key Concept: Decompose the numerator as the derivative of the denominator plus a remainder term to separate the integral into a logarithmic part and a simple part.
Recognize that the integrand $\frac{(e^x+\cos x+1)-(e^x+\sin x+x)}{(e^x+\sin x+x)}$ is of the form $\frac{f(x)+g(x)}{e^x+\sin x+x}$ where $f(x) = e^x+\sin x+x$ and $g(x) = -x$. This can be split as $\frac{f'(x)}{f(x)} + \frac{g(x)}{e^x+\sin x+x}$. The integral becomes $\ln|e^x+\sin x+x| - x + c$ since $\frac{d}{dx}(e^x+\sin x+x) = e^x+\cos x+1$.
Correct Answer: 3