若 $f(x)$ 有连续导数,且
$$
\int_0^\pi f(x) \sin x \mathrm{~d} x=k, f(\pi)=-2, f(0)=5,
$$
则 $\int_0^\pi f^{\prime}(x) \cos x \mathrm{~d} x=$
【答案】 $k-3$

【解析】
【参考解析】由被积函数表达式的结构,抽象函数的导数的乘 积,所以考虑分部积分法,于是有
$$
\begin{aligned}
& \int_0^\pi f^{\prime}(x) \cos x \mathrm{~d} x=\int_0^\pi \cos x \mathrm{~d} f(x) \\
=& {[f(x) \cos x]_0^\pi+\int_0^\pi f(x) \sin x \mathrm{~d} x } \\
=& f(\pi)(-1)-f(0)+\int_0^\pi f(x) \sin x \mathrm{~d} x \\
=& 2-5+k=k-3 .
\end{aligned}
$$
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