Which of the following must be the true value of the statements below in that order. Statement I: e a x . sin b x ? d x = e a x a 2 + b 2 ( a sin b x - b cos b x ) + c , where a , b R - { 0 } and c is an arbitrary real number. Statement II: f '

Question

$Which of the following must be the true value of the statements below in that order.$

$Statement I: \int e^{a x} . \sin b x d x = \frac{e^{a x}}{a^{2} + b^{2}} (a \sin b x - b \cos b x) + c, where a, b \in R - {0} and c is an arbitrary real number.$

$Statement II: \int \frac{f' (x)}{(f {(x))}^{n}} d x = \frac{(f {(x))}^{- n + 1}}{- n + 1}, where n \neq 1 and f (x) is a positive differentiable function.$

$Statement III: \int e^{k x} \frac{f (k x) + f' (k x)}{k} d x = e^{k x} f (k x) + c, where k, c \in R .$

$Statement IV: Let F (x) be an indefinite integral of \sin^{2} x, then the function F (x) satisfies$ $F (x + π) = F (x) for all real x .$

Smart Achievers · Accepted Answer

(4)

$Statement-I: Use by parts 2 times$

$Statement-II: Indefinite integral + c must be kept$

$Statement-IV: F (x) = \int \sin^{2} x d x = g (x) + c$

$F (x + π) = F (x) might not satisfy for every' c', (also integral contains linear expression in x)$

Solution

1 TFFT

2 TFTF

3 TFTT

4 TFFF

Ans.
(4)

$Statement-I: Use by parts 2 times$

$Statement-II: Indefinite integral + c must be kept$

$Statement-IV: F (x) = \int \sin^{2} x d x = g (x) + c$

$F (x + π) = F (x) might not satisfy for every' c', (also integral contains linear expression in x)$

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