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Complete ∫ex(1x+lnx)dx\int_{ }^{ }e^x\left(\frac{1}{x}+\ln x\right)dx∫ex(x1+lnx)dx =
ex−ce^x-cex−c
12x2(lnx−12)+c\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c21x2(lnx−21)+c
exlnlnx+ce^x\ln\ln x+cexlnlnx+c
Complete ∫x3cosxdx\int_{ }^{ }x^3\cos xdx∫x3cosxdx =
tanx−c\tan x-ctanx−c
ln∣secx+tanx∣\ln\left|\sec x+\tan x\right|ln∣secx+tanx∣
x3sinx+3x2cosx−6xsinx−6cosx+cx^3\sin x+3x^2\cos x-6x\sin x-6\cos x+cx3sinx+3x2cosx−6xsinx−6cosx+c
Fill in the blank:∫x4.lnxdx=..........\int_{ }^{ }x^4.\ln xdx=..........∫x4.lnxdx=..........
x55(lnx−15)+c\frac{x^5}{5}\left(\ln x-\frac{1}{5}\right)+c5x5(lnx−51)+c
xlnx−x+cx\ln x-x+cxlnx−x+c
x33(lnx−13)+c\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c3x3(lnx−31)+c
Complete ∫sin−1xdx=\int_{ }^{ }\sin^{-1}xdx=∫sin−1xdx=
xsin−1x−1−x2+cx\sin^{-1}x-\sqrt{1-x^2}+cxsin−1x−1−x2+c
Complete ∫t2.lntdt =\int_{ }^{ }t^2.\ln tdt\ =∫t2.lntdt =
12t2(lnt−12)+c\frac{1}{2}t^2\left(\ln t-\frac{1}{2}\right)+c21t2(lnt−21)+c
tlnt−t+ct\ln t-t+ctlnt−t+c
t33(lnt−13)+c\frac{t^3}{3}\left(\ln t-\frac{1}{3}\right)+c3t3(lnt−31)+c
complete ∫e−x(cosx−sinx)dx=\int_{ }^{ }e^{-x}\left(\cos x-\sin x\right)dx=∫e−x(cosx−sinx)dx=
x44(e−xsinx+c−14)+c\frac{x^4}{4}\left(e^{-x}\sin x+c-\frac{1}{4}\right)+c4x4(e−xsinx+c−41)+c
e−xsinx+ce^{-x}\sin x+ce−xsinx+c
x33(e−xsinx+c−13)+c\frac{x^3}{3}\left(e^{-x}\sin x+c-\frac{1}{3}\right)+c3x3(e−xsinx+c−31)+c
Complete ∫e3x(2sinx−cosx)xdx\int_{ }^{ }\frac{e^{3x}\left(2\sin x-\cos x\right)}{x}dx∫xe3x(2sinx−cosx)dx =
e3x+cosx+ce^{3x}+\cos x+ce3x+cosx+c
e3xln∣secx+tanx∣e^{3x}\ln\left|\sec x+\tan x\right|e3xln∣secx+tanx∣
e3xcscx+ce^{3x}\csc x+ce3xcscx+c
Complete ∫x.ex(1+x)2dx\int_{ }^{ }\frac{x.e^x}{\left(1+x\right)^2}dx∫(1+x)2x.exdx =
ex1+x+c\frac{e^x}{1+x}+c1+xex+c
e2x+ce^{2x}+ce2x+c
ln∣sinx∣+c\ln\left|\sin x\right|+cln∣sinx∣+c
complete ∫x(Tan−1x)dx\int_{ }^{ }x\left(Tan^{-1}x\right)dx∫x(Tan−1x)dx =
tanx-c
12tanx−1(x2+1)−12x+c\frac{1}{2}\tan x^{-1}\left(x^2+1\right)-\frac{1}{2}x+c21tanx−1(x2+1)−21x+c
sinx−cosx+c\sin x-\cos x+csinx−cosx+c
Fill in the blank:∫lnxdx=.......\int_{ }^{ }\ln xdx=.......∫lnxdx=.......
It is done.