\int_{0}^{\pi/3 }{(f'(3x)+f'''(3x))}cos3xdx=-5 find f"(0) +f"(\pi )
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3 Answers
We have
-5=\int_0^{\pi/3}f^\prime(3x)\cos 3x\ \mathrm{d}x + \int_0^{\pi/3}f^{\prime \prime\prime}(3x)\cos 3x\ \mathrm{d}x=I_1+I_2
Next,
I_1=\int_0^{\pi/3}f^\prime(3x)\cos 3x\ \mathrm{d}x =\left|f^\prime(3x)\dfrac{\sin3x}{3}\right|_0^{\pi/3}-\int_0^{\pi/3}3f^{\prime\prime}(3x) \dfrac{\sin3x}{3}\ \mathrm{d}x
And
I_2=\int_0^{\pi/3}f^{\prime\prime\prime}(3x)\cos 3x\ \mathrm{d}x =\left|\dfrac{f^{\prime\prime}(3x)}{3}\,\cos 3x\right|_0^{\pi/3}+\int_0^{\pi/3}3 \sin3x\,\dfrac{f^{\prime\prime}(3x)}{3}\ \mathrm{d}x
Adding them we get
I_1+I_2 = -\dfrac{1}{3}f^{\prime\prime}(\pi)-\dfrac{1}{3}f^{\prime\prime}(0)
which give us
15 = f^{\prime\prime}(\pi)+f^{\prime\prime}(0)