Cho tích phân 0^pi/2 f( x )dx = 5. Tính I = tích phân 0^pi/2 f( x ) + 2sin x dx = 5

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Cho \(\int\limits_0^{\frac{\pi }{2}} {f\left( x \right){\rm{d}}x} = 5\). Tính \(I = \int\limits_0^{\frac{\pi }{2}} {\left[ {f\left( x \right) + 2\sin x} \right]{\rm{d}}x} = 5\).

\(I = 7\)

\(I = 5 + \frac{\pi }{2}\)

\(I = 3\)

\(I = 5 + \pi \)

Giải thích

Chọn A

Ta có

\(I = \int\limits_0^{\frac{\pi }{2}} {\left[ {f\left( x \right)\, + 2\sin x} \right]\,{\rm{d}}x = \int\limits_0^{\frac{\pi }{2}} {f\left( x \right)\,{\rm{d}}x\,{\rm{ + 2}}\int\limits_0^{{\textstyle{\pi \over 2}}} {\sin x\,{\rm{d}}x} } } \)\( = \int\limits_0^{\frac{\pi }{2}} {f\left( x \right)} \,{\rm{d}}x\,\, - \left. {2\cos x} \right|_0^{\frac{\pi }{2}} = 5 - 2\left( {0 - 1} \right) = 7\).