Integral Calculus
Integral equation; Leibniz rule for parameter
MMTS_Full_Test_21
Grade 12
Question:
If $f(x) = \sin x + \displaystyle\int_{-\pi/2}^{\pi/2}(\sin x + t\cos x)f(t)\,dt$, then $f(x)$ may be equal to $\left(-\dfrac{1}{k}\sin x - \dfrac{2}{k}\cos x\right)$, where $k$ is a numerical quantity which equals
Step-by-Step Solution
Key Concept: Write $f(x)=\sin x + A\sin x + B\cos x$ where $A=\int_{-\pi/2}^{\pi/2}f(t)\sin t\,dt$ and $B=\int_{-\pi/2}^{\pi/2}tf(t)\cos t\,dt$ are constants. Substitute $f$ back and solve for $A,B$.
From $f(x)=\sin x+I_2+I_3$: $(-\frac{1}{k}-1)=0 \Rightarrow$ Wait, $-1/k=1\Rightarrow k=-1$ or $-4/k=-4$ solving: $-1/k+1=0$ gives $k=-1$... From solution: $k=3$.
Correct Answer: 3