If F' (X) = a Sin X + B Cos X and F' (0) = 4, F(0) = 3, F ( π 2 ) = 5, Find F(X) - Mathematics

Sum

If f' (x) = a sin x + b cos x and f' (0) = 4, f(0) = 3, f

$\left( \frac{\pi}{2} \right)$ = 5, find f(x)

Solution

$f'\left( x \right) = a \ sin x + b \cos x$
$f'\left( 0 \right) = 4, f\left( 0 \right) = 3$
$f\left( \frac{\pi}{2} \right) = 5$
$f'\left( x \right) = a \sin x + b \cos x$
$\int{f}'\left( x \right)dx = \int\left( a \sin x + b \cos x \right)dx$
$f\left( x \right) = - a \cos x + b \sin x + C . . . (i)$
$Now puting x = 0 in equation (i)$
$f\left( 0 \right) = - a \cos 0 + b \sin 0 + C$
$3 = - a \times 1 + b \times 0 + C$
$3 = - a + C . . . \left( ii \right)$
$\text{Now puting x} = \frac{\pi}{2} \text{in equation} (i)$
$f\left( \frac{\pi}{2} \right) = - a \cos \frac{\pi}{2} + b \sin \frac{\pi}{2} + C$
$5 = - a \cos\frac{\pi}{2} + b \sin \left( \frac{\pi}{2} \right) + C$
$5 = - a \times 0 + b \times 1 + C$
$5 = b + C . . . \left( iii \right)$
$\text{We also have }f'\left( 0 \right) = 4$
$f'\left( x \right) = a \sin x + b \cos x$
$f'\left( 0 \right) = a \sin 0 + b \cos 0$
$4 = a \times 0 + b \times 1$
$4 = b . . . \left( iv \right)$
$\text{solving} \left( ii \right), \left( iii \right) and \left( iv \right) \text{we get},$
$b = 4$
$C = 1$
$a = - 2$
$\therefore f\left( x \right) = 2\cos x + 4 \sin x + 1$

Concept: Indefinite Integral Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.2 | Q 48 | Page 15