∫ ( x + 4 ) 2 x + 3 e x d x is equal to

Suppose, I = ∫ ( x + 4 ) 2 x + 3 e x d x = ∫ ( x + 4 ) 2 ( x + 4 ) 1 e x d x = ∫ ( x + 4 ) e x − ∫ ( x + 4 ) 2 e x d x = ∫ e x ( ( x + 4 ) 1 − ( x + 4 ) 2 1 ) d x = e x ( x + 4 ) 1 + C [ ∫ e x { f ( x ) + f ′ ( x ) } d x = e x f ( x ) + C ]

Mathematics - Indefinite Integration Question with Solution | TestHub

MathematicsIndefinite IntegrationBy partMedium2 minPYQ_2023

MathematicsMediumsingle choice

$\int \frac{x + 3}{( x + 4 ) ^{2}} e^{x} d x$ is equal to

Options:

Answer:

Solution:

Suppose,
$I = \int \frac{x + 3}{( x + 4 ) ^{2}} e^{x} d x = \int \frac{( x + 4 ) 1}{( x + 4 ) ^{2}} e^{x} d x$
$= \int \frac{e ^{x}}{( x + 4 )} - \int \frac{e ^{x}}{( x + 4 ) ^{2}} d x$
$= \int e^{x} (\frac{1}{( x + 4 )} - \frac{1}{( x + 4 ) ^{2}}) d x$
$= e^{x} \frac{1}{( x + 4 )} + C$
$[\int e^{x} {f (x) + f^{'} (x)} d x = e^{x} f (x) + C]$

Stream:BITSATSubject:MathematicsTopic:Indefinite IntegrationSubtopic:By part

⏱ 2mℹ️ Source: PYQ_2023

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