Home
Class 12
MATHS
If int(x^2-x+1)/((x^2+1)^(3/2))e^x dx=e^...

If `int(x^2-x+1)/((x^2+1)^(3/2))e^x dx=e^xf(x)+c`, then (a) `f(x)` is an even function (b) `f(x)` is a bounded function (c) the range of `f(x)` is `(0,1)` (d) `f(x)` has two points of extrema

A

`f(x)` is an even function

B

`f(x)` is a bounded function

C

the range of `f(x)` is (0, 1]

D

`f(x)` has two points of extrema

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`I=int (x^(2)-x+1)/((x^(2)+1)^(3//2))e^(x)dx`
`=inte^(x)[(x^(2)+1)/((x^(2)+1)^(3//2))-(x)/((x^(2)+1)^(3//2))]dx`
`=int e^(x)[(1)/(sqrt(x^(2)+1))+{(-x)/((x^(2)+1)^(3//2))}]dx`
`=e^(x)[f(x)+f'(x)]dx, " where "f(x)=(1)/(sqrt(x^(2)+1))`
`=e^(x) f(x)+c=(e^(x))/(sqrt(x^(2)+1))+c`
The graph of `f(x)` is given in the figure.

From the graph, `f(x)` is even, bounded function and has the range (0, 1].
Promotional Banner

Topper's Solved these Questions

  • INDEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise Exercises (Linked Comprehension Type)|17 Videos

  • INDEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise EXERCISES (Matrix Match Type)|4 Videos

  • INDEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise EXERCISES (Single Correct Answer Type)|77 Videos

  • HYPERBOLA

    CENGAGE ENGLISH|Exercise COMPREHENSION TYPE|2 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE ENGLISH|Exercise Single correct Answer|21 Videos

Similar Questions

Explore conceptually related problems

If int((x-1)/(x^2))e^x dx=f(x)e^x+c , then write the value of f(x) .

If int((x-1)/(x^2))e^x\ dx=f(x)\ e^x+C , then write the value of f(x) .

int(x+1)/(x(1+x e^x)^2)dx=log|1-f(x)|+f(x)+c , then f(x)=

If int (e^x-1)/(e^x+1)dx=f(x)+C, then f(x) is equal to

Given inte^x\ (tanx+1)secx\ dx=e^xf(x)+c , write f(x) satisfying the above.

Let f(x) be a continuous function in R such that f(x) does not vanish for all x in R . If int_1^5 f(x)dx=int_-1^5 f(x)dx , then in R, f(x) is (A) an even function (B) an odd function (C) a periodic function with period 5 (D) none of these

If int(dx)/(x^2+a x+1)=f(g(x))+c , then f(x) is inverse trigonometric function for |a|>2 f(x) is logarithmic function for |a| 2 g(x) is rational function for |a|<2

Let f be a differential function such that f(x)=f(2-x) and g(x)=f(1 +x) then (1) g(x) is an odd function (2) g(x) is an even function (3) graph of f(x) is symmetrical about the line x= 1 (4) f'(1)=0

Let f be a differential function such that f(x)=f(2-x) and g(x)=f(1 +x) then (1) g(x) is an odd function (2) g(x) is an even function (3) graph of f(x) is symmetrical about the line x= 1 (4) f'(1)=0

If x!=1\ a n d\ f(x)=(x+1)/(x-1) is a real function, then f(f(f(2))) is (a) 1 (b) 2 (c) 3 (d) 4