考研数学：常见的初等函数求导公式以及其对应的积分公式
(xu)′=μxk−1∫μxn−1dx=xμ+c\left(x^{u}\right)^{\prime}=\mu x^{k-1} \quad \quad \int \mu x^{n-1} \mathrm{d} x=x^{\mu}+c(xu)′=μxk−1∫μxn−1dx=xμ+c(xmp)′=m−ppxmp∫mpxm−ppdx=xmp=xmp+c(\sqrt[p]{x^{m}})^{\prime}=
(xu)′=μxk−1∫μxn−1dx=xμ+c \left(x^{u}\right)^{\prime}=\mu x^{k-1} \quad \quad \int \mu x^{n-1} \mathrm{d} x=x^{\mu}+c (xu)′=μxk−1∫μxn−1dx=xμ+c
(xmp)′=m−ppxmp∫mpxm−ppdx=xmp=xmp+c (\sqrt[p]{x^{m}})^{\prime}=\frac{m-p}{p} x^{\frac{m}{p}} \quad \int \frac{m}{p} x^{\frac{m-p}{p}} \mathrm{d} x=x^{\frac{m}{p}}=\sqrt[p]{x^{m}}+c (pxm)′=pm−pxpm∫pmxpm−pdx=xpm=pxm+c
(ln∣x∣)′=1x∫1xdx=ln∣x∣+c (\ln |x|)^{\prime}=\frac{1}{x} \quad \int \frac{1}{x} \mathrm{d} x=\ln |x|+c (ln∣x∣)′=x1∫x1dx=ln∣x∣+c
(ex)′=ex∫exdx=ex+c \left(\mathrm{e}^{x}\right)^{\prime}=\mathrm{e}^{x} \quad \int \mathrm{e}^{x} \mathrm{d} x=\mathrm{e}^{x}+c (ex)′=ex∫exdx=ex+c
(ax)′=axlna∫axdx=axlna+c \left(a^{x}\right)^{\prime}=a^{x} \ln a \quad \int a^{x} \mathrm{d} x=\frac{a^{x}}{\ln a}+c (ax)′=axlna∫axdx=lnaax+c
(sinx)′=cosx∫cosxdx=sinx+c (\sin x)^{\prime}=\cos x \quad \int \cos x d x=\sin x+c (sinx)′=cosx∫cosxdx=sinx+c
(cosx)′=−sinx∫sinxdx=−cosx+c (\cos x)^{\prime}=-\sin x \quad \int \sin x d x=-\cos x+c (cosx)′=−sinx∫sinxdx=−cosx+c
(tanx)′=sec2x∫sec2xdx=tanx+c (\tan x)^{\prime}=\sec ^{2} x \quad \int \sec ^{2} x d x=\tan x+c (tanx)′=sec2x∫sec2xdx=tanx+c
(cotx)′=−csc2x∫csc2xdx=−cotx+c (\cot x)^{\prime}=-\csc ^{2} x \quad \int \csc ^{2} x d x=-\cot x+c (cotx)′=−csc2x∫csc2xdx=−cotx+c
(secx)′=secxtanx∫secxtanxdx=secx+c (\sec x)^{\prime}=\sec x \tan x \quad \int \sec x \tan x d x=\sec x+c (secx)′=secxtanx∫secxtanxdx=secx+c
(cscx)′=−cscxcotx∫cscxcotxdx=−cscx+c (\csc x)^{\prime}=-\csc x \cot x \quad \int \csc x \cot x d x=-\csc x+c (cscx)′=−cscxcotx∫cscxcotxdx=−cscx+c
(arcsinx)′=11−x2∫11−x2dx=arcsinx+c (\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}} \quad \int \frac{1}{\sqrt{1-x^{2}}} \mathrm{d} x=\arcsin x+c (arcsinx)′=1−x21∫1−x21dx=arcsinx+c
(arccosx)′=−11−x2∫11−x2dx=−arccosx+c (\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}} \quad \int \frac{1}{\sqrt{1-x^{2}}} \mathrm{d} x=-\arccos x+c (arccosx)′=−1−x21∫1−x21dx=−arccosx+c
(arctanx)′=11+x2∫11+x2dx=arctanx+c (\arctan x)^{\prime}=\frac{1}{1+x^{2}} \quad \int \frac{1}{1+x^{2}} \mathrm{d} x=\arctan x+c (arctanx)′=1+x21∫1+x21dx=arctanx+c
(arccotx)′=−11+x2∫11+x2dx=−arccotx+c (\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}} \quad \int \frac{1}{1+x^{2}} \mathrm{d} x=-\operatorname{arccot} x+c (arccotx)′=−1+x21∫1+x21dx=−arccotx+c
(arcsecx)′=1xx2−1∫1xx2−1dx=arcsecx+c (\operatorname{arcsec} x)^{\prime}=\frac{1}{x \sqrt{x^{2}-1}} \quad \int \frac{1}{x \sqrt{x^{2}-1}} \mathrm{d} x=\operatorname{arcsec} x+c (arcsecx)′=xx2−11∫xx2−11dx=arcsecx+c
(arccscx)′=−1xx2−1∫1xx2−1dx=−arccscx+c (\operatorname{arccsc} x)^{\prime}=-\frac{1}{x \sqrt{x^{2}-1}} \quad \int \frac{1}{x \sqrt{x^{2}-1}} \mathrm{d} x=-\operatorname{arccsc} x+c (arccscx)′=−xx2−11∫xx2−11dx=−arccscx+c
(ln∣x+x2±b∣)′=1x2±b∫1x2±bdx=ln∣x+x2±b∣+c (\ln |x+\sqrt{x^{2} \pm b}|)^{\prime}=\frac{1}{\sqrt{x^{2} \pm b}} \quad \int \frac{1}{\sqrt{x^{2} \pm b}} \mathrm{d} x=\ln |x+\sqrt{x^{2} \pm b}|+c (ln∣x+x2±b∣)′=x2±b1∫x2±b1dx=ln∣x+x2±b∣+c
补充：几个特殊的三角函数积分公式
∫sin2xdx=x2−sin2x4+C(sin2x=1−cos2x2) \int \sin ^{2} x \mathrm{d} x=\frac{x}{2}-\frac{\sin 2 x}{4}+C\left(\sin ^{2} x=\frac{1-\cos 2 x}{2}\right) ∫sin2xdx=2x−4sin2x+C(sin2x=21−cos2x)
∫cos2xdx=x2+sin2x4+C(cos2x=1+cos2x2) \int \cos ^{2} x \mathrm{d} x=\frac{x}{2}+\frac{\sin 2 x}{4}+C\left(\cos ^{2} x=\frac{1+\cos 2 x}{2}\right) ∫cos2xdx=2x+4sin2x+C(cos2x=21+cos2x)
∫tan2xdx=tanx−x+C(tan2x=sec2x−1) \int \tan ^{2} x \mathrm{d} x=\tan x-x+C\left(\tan ^{2} x=\sec ^{2} x-1\right) ∫tan2xdx=tanx−x+C(tan2x=sec2x−1)
∫cot2xdx=−cotx−x+C(cot2x=csc2x−1) \int \cot ^{2} x \mathrm{d} x=-\cot x-x+C\left(\cot ^{2} x=\csc ^{2} x-1\right) ∫cot2xdx=−cotx−x+C(cot2x=csc2x−1)
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