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Finding integrals by integration by parts

Complete $\int_{ }^{ }\frac{x.e^x}{\left(1+x\right)^2}dx$ =

$\frac{e^x}{1+x}+c$

$e^{2x}+c$

$\ln\left|\sin x\right|+c$

Fill in the blank: $\int_{ }^{ }x^4.\ln xdx=..........$

$\frac{x^5}{5}\left(\ln x-\frac{1}{5}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$

Complete $\int_{ }^{ }e^x\left(\frac{1}{x}+\ln x\right)dx$ =

$e^x-c$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$e^x\ln\ln x+c$

Fill in the blank: $\int_{ }^{ }\ln xdx=.......$

$\tan x-c$

$\sin x-\cos x+c$

$x\ln x-x+c$

Fill in the blank: $\int_{ }^{ }x\ln xdx=........$

$\tan x-c$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$x\ln x-x+c$

complete $\int_{ }^{ }e^{-x}\left(\cos x-\sin x\right)dx=$

$\frac{x^4}{4}\left(e^{-x}\sin x+c-\frac{1}{4}\right)+c$

$e^{-x}\sin x+c$

$\frac{x^3}{3}\left(e^{-x}\sin x+c-\frac{1}{3}\right)+c$

Fill in the blank: $\int_{ }^{ }xe^xdx=........$

$xe^x-e^x+c$

$e^{2x}+cl$

$n\left|\sin x\right|+c$

Complete $\int_{ }^{ }e^x\left(\sin x+\cos x\right)dx$ =

$-x^2\cos x$

$e^x\ln\left|\sec x+\tan x\right|$

$e^x\sin x+c$

Fill in the blank: $\int_{ }^{ }x^2\ln xdx=.........$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$

Fill in the blank: $\int_{ }^{ }x^3\ln xdx=.......$

$\frac{x^4}{4}\left(\ln x-\frac{1}{\left(4\right)}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$

Questions

Time

Score

Finding integrals by integration by parts

Complete ﻿∫x.ex(1+x)2dx\int_{ }^{ }\frac{x.e^x}{\left(1+x\right)^2}dx∫​(1+x)2x.ex​dx﻿ =

﻿ex1+x+c\frac{e^x}{1+x}+c1+xex​+c﻿

﻿e2x+ce^{2x}+ce2x+c﻿

﻿ln⁡∣sin⁡x∣+c\ln\left|\sin x\right|+cln∣sinx∣+c﻿

Fill in the blank:﻿∫x4.ln⁡xdx=..........\int_{ }^{ }x^4.\ln xdx=..........∫​x4.lnxdx=..........﻿

﻿x55(ln⁡x−15)+c\frac{x^5}{5}\left(\ln x-\frac{1}{5}\right)+c5x5​(lnx−51​)+c﻿

﻿xln⁡x−x+cx\ln x-x+cxlnx−x+c﻿

﻿x33(ln⁡x−13)+c\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c3x3​(lnx−31​)+c﻿

Complete ﻿∫ex(1x+ln⁡x)dx\int_{ }^{ }e^x\left(\frac{1}{x}+\ln x\right)dx∫​ex(x1​+lnx)dx﻿ =

﻿ex−ce^x-cex−c﻿

﻿12x2(ln⁡x−12)+c\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c21​x2(lnx−21​)+c﻿

﻿exln⁡ln⁡x+ce^x\ln\ln x+cexlnlnx+c﻿

Fill in the blank:﻿∫ln⁡xdx=.......\int_{ }^{ }\ln xdx=.......∫​lnxdx=.......﻿

﻿tan⁡x−c\tan x-ctanx−c﻿

﻿sin⁡x−cos⁡x+c\sin x-\cos x+csinx−cosx+c﻿

﻿xln⁡x−x+cx\ln x-x+cxlnx−x+c﻿

Fill in the blank:﻿∫xln⁡xdx=........\int_{ }^{ }x\ln xdx=........∫​xlnxdx=........﻿

﻿tan⁡x−c\tan x-ctanx−c﻿

﻿12x2(ln⁡x−12)+c\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c21​x2(lnx−21​)+c﻿

﻿xln⁡x−x+cx\ln x-x+cxlnx−x+c﻿

complete ﻿∫e−x(cos⁡x−sin⁡x)dx=\int_{ }^{ }e^{-x}\left(\cos x-\sin x\right)dx=∫​e−x(cosx−sinx)dx=﻿

﻿x44(e−xsin⁡x+c−14)+c\frac{x^4}{4}\left(e^{-x}\sin x+c-\frac{1}{4}\right)+c4x4​(e−xsinx+c−41​)+c﻿

﻿e−xsin⁡x+ce^{-x}\sin x+ce−xsinx+c﻿

﻿x33(e−xsin⁡x+c−13)+c\frac{x^3}{3}\left(e^{-x}\sin x+c-\frac{1}{3}\right)+c3x3​(e−xsinx+c−31​)+c﻿

Fill in the blank:﻿∫xexdx=........\int_{ }^{ }xe^xdx=........∫​xexdx=........﻿

﻿xex−ex+cxe^x-e^x+cxex−ex+c﻿

﻿e2x+cle^{2x}+cle2x+cl﻿

﻿n∣sin⁡x∣+cn\left|\sin x\right|+cn∣sinx∣+c﻿

Complete ﻿∫ex(sin⁡x+cos⁡x)dx\int_{ }^{ }e^x\left(\sin x+\cos x\right)dx∫​ex(sinx+cosx)dx﻿ =

﻿−x2cos⁡x-x^2\cos x−x2cosx﻿

﻿exln⁡∣sec⁡x+tan⁡x∣e^x\ln\left|\sec x+\tan x\right|exln∣secx+tanx∣﻿

﻿exsin⁡x+ce^x\sin x+cexsinx+c﻿

Fill in the blank:﻿∫x2ln⁡xdx=.........\int_{ }^{ }x^2\ln xdx=.........∫​x2lnxdx=.........﻿

﻿12x2(ln⁡x−12)+c\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c21​x2(lnx−21​)+c﻿

﻿xln⁡x−x+cx\ln x-x+cxlnx−x+c﻿

﻿x33(ln⁡x−13)+c\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c3x3​(lnx−31​)+c﻿

Fill in the blank:﻿∫x3ln⁡xdx=.......\int_{ }^{ }x^3\ln xdx=.......∫​x3lnxdx=.......﻿

﻿x44(ln⁡x−1(4))+c\frac{x^4}{4}\left(\ln x-\frac{1}{\left(4\right)}\right)+c4x4​(lnx−(4)1​)+c﻿

﻿xln⁡x−x+cx\ln x-x+cxlnx−x+c﻿

﻿x33(ln⁡x−13)+c\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c3x3​(lnx−31​)+c﻿

Incorrect

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Complete $\int_{ }^{ }\frac{x.e^x}{\left(1+x\right)^2}dx$ =

$\frac{e^x}{1+x}+c$

$e^{2x}+c$

$\ln\left|\sin x\right|+c$

Fill in the blank: $\int_{ }^{ }x^4.\ln xdx=..........$

$\frac{x^5}{5}\left(\ln x-\frac{1}{5}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$

Complete $\int_{ }^{ }e^x\left(\frac{1}{x}+\ln x\right)dx$ =

$e^x-c$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$e^x\ln\ln x+c$

Fill in the blank: $\int_{ }^{ }\ln xdx=.......$

$\tan x-c$

$\sin x-\cos x+c$

$x\ln x-x+c$

Fill in the blank: $\int_{ }^{ }x\ln xdx=........$

$\tan x-c$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$x\ln x-x+c$

complete $\int_{ }^{ }e^{-x}\left(\cos x-\sin x\right)dx=$

$\frac{x^4}{4}\left(e^{-x}\sin x+c-\frac{1}{4}\right)+c$

$e^{-x}\sin x+c$

$\frac{x^3}{3}\left(e^{-x}\sin x+c-\frac{1}{3}\right)+c$

Fill in the blank: $\int_{ }^{ }xe^xdx=........$

$xe^x-e^x+c$

$e^{2x}+cl$

$n\left|\sin x\right|+c$

Complete $\int_{ }^{ }e^x\left(\sin x+\cos x\right)dx$ =

$-x^2\cos x$

$e^x\ln\left|\sec x+\tan x\right|$

$e^x\sin x+c$

Fill in the blank: $\int_{ }^{ }x^2\ln xdx=.........$

$\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$

Fill in the blank: $\int_{ }^{ }x^3\ln xdx=.......$

$\frac{x^4}{4}\left(\ln x-\frac{1}{\left(4\right)}\right)+c$

$x\ln x-x+c$

$\frac{x^3}{3}\left(\ln x-\frac{1}{3}\right)+c$