if \(∫ \frac{(x^2+1)e^x}{(x+1)^2}dx = ƒ(x)e^x+C\) where \(C\) is a constant, then \(\frac{d^3ƒ}{dx^3}\) at \(x = 1\) is equal to :
- \(-\frac{3}{4}\)
- \(\frac{3}{4}\)
- \(-\frac{3}{2}\)
- \(\frac{3}{2}\)
The Correct Option is B
Solution and Explanation
\(I = ∫ \frac{(x^2+1)e^x}{(x+1)^2}dx = ƒ(x)e^x+C\)
\(I = ∫ \frac{e^x(x^2-1+1+1)}{(x+1)^2}dx\)
= \(∫ e^x\bigg[\frac{x-1}{x+1}+\frac{2}{(x+1)^2}\bigg]dx\)
= \(e^x\bigg(\frac{x-1}{x+1}\bigg)+c\)
\(∴ f (x) = \frac{x-1}{x+1}\)
\(f(x) = \frac{1- 2}{x+1}\)
\(f' (x) = 2\bigg(\frac{1}{x+1}\bigg)^2\)
\(f''(x) = -4\bigg(\frac{1}{x+1}\bigg)^3\)
\(f'''(x) = \frac{12}{(x+1)^4}\)
for \(x = 1\)
\(f'''(1) = \frac{12}{24}\)
= \(\frac{12}{16}\)
= \(\frac{3}{4}\)
Hence, the correct option is (B): \(\frac{3}{4}\)
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Concepts Used:
Applications of Integrals
There are distinct applications of integrals, out of which some are as follows:
In Maths
Integrals are used to find:
- The center of mass (centroid) of an area having curved sides
- The area between two curves and the area under a curve
- The curve's average value
In Physics
Integrals are used to find:
- Centre of gravity
- Mass and momentum of inertia of vehicles, satellites, and a tower
- The center of mass
- The velocity and the trajectory of a satellite at the time of placing it in orbit
- Thrust