Question

If P = $\int_{}^{}e^{ax}\cos$ bx dx and Q = $\int_{}^{}e^{ax}\sin$bx dx, then (P² + Q²) is equal to (neglect integration constant)

• A. $\frac{e^{ax}}{a^{2} + b^{2}}$
• B. $\frac{e^{bx}}{a^{2} + b^{2}}$
• C. $\frac{e^{2ax}}{a^{2} + b^{2}}$
• D. $\frac{e^{(a + b)x}}{a^{2} + b^{2}}$

Answer 1

Answer: (C)

$\frac{e^{2ax}}{a^{2} + b^{2}}$

Answer 2

P + iQ = $\int_{}^{}{e^{ax}\cos bxdx}$ + i $\int_{}^{}{e^{ax}\sin bxdx}$

P + iQ = $\int_{}^{}{e^{ax}(\cos bx + i\sin bx)}$ dx

P + iQ = $\int_{}^{}{(e^{ax}e^{ibx})}$ dx

P + iQ = $\int_{}^{}e^{(ax + ibx)}$ dx

P + iQ = $\frac{e^{(a + ib)x}}{(a + ib)}$

(P + iQ) = $\frac{e^{ax}e^{ibx}}{(a + ib)}$

Take modulus on both side

$\sqrt { \mathrm { P } ^ { 2 } + \mathrm { Q } ^ { 2 } }$ = $\frac{e^{ax}}{\sqrt{a^{2} + b^{2}}}$

P² + Q² = $\frac{e^{2ax}}{(a^{2} + b^{2})}$

Q = cos bx + i sin bx

= $\sqrt{\cos^{2}bx + \sin^{2}bx}$ = 1