Question:
The area bounded by the curves y = f (x), the X-axis and
the ordinates x = 1 and x = b is (b - 1 ) sin (3b + 4). Then,
f (x) is equal to
The area bounded by the curves y = f (x), the X-axis and
the ordinates x = 1 and x = b is (b - 1 ) sin (3b + 4). Then,
f (x) is equal to
Updated On: June 02, 2025
- (a) (x - 1) cos (3x + 4)
- (b) 8sin (3x + 4)
- (c) sin (3x + 4) + 3(x - 1) cos (3x + 4)
- (d) None of the above
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The Correct Option is C
Solution and Explanation
Since, $ \int \limits_1^b f(x) dx = (b - 1) sin (3b + 4) $
On differentiating both sides w.r.t. b, we get
$ f(b) = 3(b - 1). cos(3b + 4) + sin(3b + 4) $
$ \therefore f(x) = sin(3x + 4) + 3(x - 1) cos(3x + 4) $
On differentiating both sides w.r.t. b, we get
$ f(b) = 3(b - 1). cos(3b + 4) + sin(3b + 4) $
$ \therefore f(x) = sin(3x + 4) + 3(x - 1) cos(3x + 4) $
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
