Question

If $\int e^{3x}[3\sin 4x+4\cos 4x]dx=\frac{e^{3x}}{k}\cdot f\left(x\right)+C,$ then which of the following is/are true

• A. $k=25$
• B. $k=1$
• C. $f\left(\frac{\pi }{4}\right)=0$
• D. $f\left(\frac{\pi }{4}\right)=\frac{3}{25}$

Answer 1

Answer: (B), (C)

$f\left(\frac{\pi }{4}\right)=0$

$I=\int e^{3x}[3\sin 4x+4\cos 4x]dx=3\int e^{3x}\sin 4xdx+4\int e^{3x}\cdot \cos 4xdx\{\because \int e^{ax}\sin bxdx=\frac{e^{ax}}{a^2+b^2}[a\sin bx-b\cos bx]+C\int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}[a\cos bx+b\sin bx]+C\}=\frac{3e^{3x}}{3^2+4^2}[3\sin 4x-4\cos 4x]+\frac{4e^{3x}}{3^2+4^2}[3\cos 4x+4\sin 4x]+C=\frac{e^{3x}}{3^2+4^2}[9\sin 4x-12\cos 4x+12\cos 4x+16\sin 4x]+C=\frac{e^{3x}}{25}\left[25\sin 4x\right]+C=e^{3x}\cdot \sin 4x+C\Rightarrow f\left(x\right)=\sin 4x\Rightarrow f\left(\frac{\pi }{4}\right)=\sin \pi =0$ and $k=1$