微积分/积分技巧/分部积分/解答

1. $\int -4\ln(x)dx$

令

${\begin{array}{ll}u=\ln(x)&\qquad du={\frac {dx}{x}}\\v=-4x&\qquad dv=-4dx\end{array}}$

则

${\begin{aligned}\int -4\ln(x)dx&=-4x\ln(x)-\int -4dx\\&=\mathbf {-4x\ln(x)+4x+C} \end{aligned}}$

2. $\int (-7x+38)\cos(x)dx$

令

${\begin{array}{ll}u=-7x+38&\qquad du=-7dx\\v=\sin(x)&\qquad dv=\cos(x)dx\end{array}}$

则

${\begin{aligned}\int (-7x+38)\cos(x)dx&=(-7x+38)\sin(x)+7\int \sin(x)dx\\&=\mathbf {(-7x+38)\sin(x)-7\cos(x)+C} \end{aligned}}$

3. $\int \limits _{0}^{\frac {\pi }{2}}(-6x+45)\cos(x)dx$

令

${\begin{array}{ll}u=-6x+45&\qquad du=-6dx\\v=\sin(x)&\qquad dv=\cos(x)dx\end{array}}$

则

${\begin{aligned}\int \limits _{0}^{\frac {\pi }{2}}(-6x+45)\cos(x)dx&=((-6x+45)\sin(x)){\bigg |}_{0}^{\frac {\pi }{2}}+6\int \limits _{0}^{\frac {\pi }{2}}\sin(x)dx\\&=(-3\pi +45)-6\cos(x){\bigg |}_{0}^{\frac {\pi }{2}}\\&=(-3\pi +45)+6\\&=\mathbf {-3\pi +51} \end{aligned}}$

4. $\int (5x+1)(x-6)^{4}dx$