Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
If $A = \int e^{ax}\cos bx dx$ and $B = \int e^{ax}\sin bx dx$, then which of the following may be correct?
$\left(A^2 + B^2\right)\left(a^2 + b^2\right) = e^{2ax}$
$\tan^{-1}\frac{B}{A} + \tan^{-1}\frac{b}{a} = bx$
(A) and (B) both
None of these
Step-by-Step Solution
Key Concept: Convert the complex exponential integral into rectangular form by separating modulus and argument using Euler's formula and De Moivre's theorem.
Given $A + Bi = \int a^x e^{ibx}dx = \int e^{(a+ib)x}dx = \frac{e^{(a+ib)x}}{a+ib}$, comparing moduli on both sides gives $\sqrt{A^2 + B^2}\sqrt{a^2 + b^2} = e^{ax}$. Squaring yields $(A^2 + B^2)(a^2 + b^2) = e^{2ax}$. Comparing arguments of both sides in equation (1) gives $\tan^{-1}\frac{B}{A} + \tan^{-1}\frac{b}{a} = bx$.
Correct Answer: 1,2,3