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Question
The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is (b −1) sin (3b + 4). Then, f (x) is __________ .
Options
(x − 1) cos (3x + 4)
sin (3x + 4)
sin (3x + 4) + 3 (x − 1) cos (3x +4)
none of these
Solution
sin (3x + 4) + 3 (x − 1) cos (3x +4)
\[y = f\left( x \right)\]
If A is the the area bound by the curve , x - axis , x = 1 and x = b
\[ \Rightarrow \int_1^b f\left( x \right) dx = \left[ A \right]_1^b = \left( b - 1 \right)\sin \left( 3b + 4 \right) .............\left\{\text{Given }\right\}\]
\[ \Rightarrow f\left( x \right) = \frac{d}{dx}\left( \left( x - 1 \right)\sin\left( 3x + 4 \right) \right)\]
\[ = \sin\left( 3x + 4 \right)\frac{d}{dx}\left( x - 1 \right) + \left( x - 1 \right)\frac{d}{dx}\sin\left( 3x + 4 \right)\]
\[ = \sin\left( 3x + 4 \right) + 3\left( x - 1 \right)\cos\left( 3x + 4 \right)\]
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