Integral Calculus Question 470
Question: If $ u=\int_{{}}^{{}}{e^{ax}\cos bx\ dx} $ and $ v=\int_{{}}^{{}}{e^{ax}\sin bx\ dx} $ , then $ (a^{2}+b^{2})(u^{2}+v^{2})= $
Options:
A) $ 2e^{ax} $
B) $ (a^{2}+b^{2})e^{2ax} $
C) $ e^{2ax} $
D) $ (a^{2}-b^{2})e^{2ax} $
Show Answer
Answer:
Correct Answer: C
Solution:
$ u=\int_{{}}^{{}}{e^{ax}\cos bx,dx} $ $ =e^{ax}\frac{\sin bx}{b}-\frac{a}{b}\int_{{}}^{{}}{e^{ax}.\sin bx,dx} $
$ =\frac{e^{ax}\sin bx}{b}-\frac{a}{b}v $
$ \Rightarrow bu+av=e^{ax}\sin bx $ ..?(i)
Similarly $ bv-au=-e^{ax}\cos bx $ ..?(ii)
Squaring (i) and (ii) and adding, we get
$ (a^{2}+b^{2})(u^{2}+v^{2})=e^{2ax} $ .
BETA
