If $\int\left(e^{ax}+bx\right)dx=4e^{4x}+\dfrac{3x^{2}}{2}$, find the values of $a$ and $b$.

CBSE Class 12 · 2008 · Indefinite IntegralsVery Short Answer

\int (e^{a x} + b x) d x = 4 e^{4 x} + \frac{3 x ^{2}}{2}

, find the values of

a

and

b

Answer: $a = 4, b = 3$

Differentiate both sides with respect to

x

(the derivative of an antiderivative returns the integrand):

e^{a x} + b x = \frac{d}{d x} (4 e^{4 x} + \frac{3 x ^{2}}{2}) = 16 e^{4 x} + 3 x .

Match the exponential factor and the coefficient of

x

on both sides:

e^{a x} = e^{4 x} \Rightarrow a = 4, b x = 3 x \Rightarrow b = 3.

a = 4, b = 3

CBSE Class 12 Board · 2008 · Subjective