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Calculate antiderivatives of a rational fraction

Evaluate the integral $\int_{ }^{ }\frac{2x^2}{x+1}dx$

$x^2-2x\ln\left|4x\right|+c$

$4\ln\left|x\right|+c$

$x^2-2x+2\ln\left|x+1\right|+c$

Complete this statement:
If f (x)=2Cos x and g(x)=e^x then ∫[f(x) - g(x)] dx = ___________.

$\frac{x^2}{2}+e^x+c$

$2\sin x-e^x+c$

$2e^x+x+c$

Find the antiderivative of $\int_{ }^{ }\frac{1}{x^2+4x+8}dx$

$\frac{1}{2}\frac{x+2}{4}$

$\ln\left|x+1\right|+\frac{1}{x+1}+c$

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

Evaluate the integral $\int_{ }^{ }\frac{x+1}{x-2}dx$

$\left(x+3\right)\ln\left|x+1\right|+c$

$\left(x-3\right)\ln\left|x+1\right|+c$

$x+\ln\left|x-2\right|+c$

Complete this statement:
If f (x)=Cos x and g(x)=x³ then ∫[f(x) - g(x)] dx = ___________.

$\frac{x^2}{2}+\sin x+c$

$\sin x+\frac{1}{4}x^4+c$

$\sin x-\frac{1}{4}x^4+c$

Find the antiderivative of $\int_{ }^{ }\frac{x}{x^2+2x+1}dx$

$\ln\left|x+1\right|+c$

$\ln\left|x+1\right|+\frac{1}{x+1}+c$

$-\frac{1}{9x^9}+c$

Complete this statement:
If f (x)=e^x and g(x)=x³ then ∫[f(x) - g(x)] dx =___________.

$\frac{x^2}{2}+e^x+c$

$e^x-\frac{1}{4}x^4+c$

$e^x+\frac{1}{4}x^4+c$

Find the antiderivative of $\int_{ }^{ }\frac{2x}{x^2+2}dx$

$\ln\left|x^2+2\right|+\frac{1}{x}+c$

$\ln\left|x^2+2\right|+c$

$\ln\left|\frac{x^{ }+1}{x}\right|+c$

Find the antiderivative of $\int_{ }^{ }\frac{\left(1-2x\right)}{x^2\left(x+1\right)}dx$

$\ln\left|\frac{x+1\ }{x}\right|+\frac{1}{x}+c$

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

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

Complete this statement:
If f (x)=Cos x and g(x)=loglogx then ∫[f(x) -g(x)] dx = ___________.

$2x^2\cos x-x\log\log x+c$

$\sin x+x\log\log x-x+c$

$\cos x-x\log\log x+x+c$

Questions

Time

Score

Calculate antiderivatives of a rational fraction

Evaluate the integral﻿∫2x2x+1dx\int_{ }^{ }\frac{2x^2}{x+1}dx∫​x+12x2​dx﻿

﻿x2−2xln⁡∣4x∣+cx^2-2x\ln\left|4x\right|+cx2−2xln∣4x∣+c﻿

﻿4ln⁡∣x∣+c4\ln\left|x\right|+c4ln∣x∣+c﻿

﻿x2−2x+2ln⁡∣x+1∣+cx^2-2x+2\ln\left|x+1\right|+cx2−2x+2ln∣x+1∣+c﻿

Complete this statement: If f (x)=2Cos x and g(x)=ex then ∫[f(x) - g(x)] dx = ___________.

﻿x22+ex+c\frac{x^2}{2}+e^x+c2x2​+ex+c﻿

﻿2sin⁡x−ex+c2\sin x-e^x+c2sinx−ex+c﻿

﻿2ex+x+c2e^x+x+c2ex+x+c﻿

Find the antiderivative of ﻿∫1x2+4x+8dx\int_{ }^{ }\frac{1}{x^2+4x+8}dx∫​x2+4x+81​dx﻿

﻿12x+24\frac{1}{2}\frac{x+2}{4}21​4x+2​﻿

﻿ln⁡∣x+1∣+1x+1+c\ln\left|x+1\right|+\frac{1}{x+1}+cln∣x+1∣+x+11​+c﻿

﻿12 (x+2)2+c\frac{1}{2}\ \frac{\left(x+2\right)}{2}+c21​ 2(x+2)​+c﻿

Evaluate the integral﻿∫x+1x−2dx\int_{ }^{ }\frac{x+1}{x-2}dx∫​x−2x+1​dx﻿

﻿(x+3)ln⁡∣x+1∣+c\left(x+3\right)\ln\left|x+1\right|+c(x+3)ln∣x+1∣+c﻿

﻿(x−3)ln⁡∣x+1∣+c\left(x-3\right)\ln\left|x+1\right|+c(x−3)ln∣x+1∣+c﻿

﻿x+ln⁡∣x−2∣+cx+\ln\left|x-2\right|+cx+ln∣x−2∣+c﻿

Complete this statement: If f (x)=Cos x and g(x)=x3 then ∫[f(x) - g(x)] dx = ___________.

﻿x22+sin⁡x+c\frac{x^2}{2}+\sin x+c2x2​+sinx+c﻿

﻿sin⁡x+14x4+c\sin x+\frac{1}{4}x^4+csinx+41​x4+c﻿

﻿sin⁡x−14x4+c\sin x-\frac{1}{4}x^4+csinx−41​x4+c﻿

Find the antiderivative of ﻿∫xx2+2x+1dx\int_{ }^{ }\frac{x}{x^2+2x+1}dx∫​x2+2x+1x​dx﻿

﻿ln⁡∣x+1∣+c\ln\left|x+1\right|+cln∣x+1∣+c﻿

﻿ln⁡∣x+1∣+1x+1+c\ln\left|x+1\right|+\frac{1}{x+1}+cln∣x+1∣+x+11​+c﻿

﻿−19x9+c-\frac{1}{9x^9}+c−9x91​+c﻿

Complete this statement: If f (x)=ex and g(x)=x3 then ∫[f(x) - g(x)] dx =___________.

﻿﻿x22+ex+c\frac{x^2}{2}+e^x+c2x2​+ex+c﻿

﻿ex−14x4+ce^x-\frac{1}{4}x^4+cex−41​x4+c﻿

﻿ex+14x4+ce^x+\frac{1}{4}x^4+cex+41​x4+c﻿

Find the antiderivative of ﻿∫2xx2+2dx\int_{ }^{ }\frac{2x}{x^2+2}dx∫​x2+22x​dx﻿

﻿ln⁡∣x2+2∣+1x+c\ln\left|x^2+2\right|+\frac{1}{x}+cln∣∣​x2+2∣∣​+x1​+c﻿

﻿ln⁡∣x2+2∣+c\ln\left|x^2+2\right|+cln∣∣​x2+2∣∣​+c﻿

﻿ln⁡∣x+1x∣+c\ln\left|\frac{x^{ }+1}{x}\right|+cln∣∣​xx+1​∣∣​+c﻿

Find the antiderivative of ﻿∫(1−2x)x2(x+1)dx\int_{ }^{ }\frac{\left(1-2x\right)}{x^2\left(x+1\right)}dx∫​x2(x+1)(1−2x)​dx﻿

﻿ln⁡∣x+1 x∣+1x+c\ln\left|\frac{x+1\ }{x}\right|+\frac{1}{x}+cln∣∣​xx+1 ​∣∣​+x1​+c﻿

﻿(1−2x)2(x−2)2+c\frac{\left(1-2x\right)}{2\left(x-2\right)^2}+c2(x−2)2(1−2x)​+c﻿

﻿ln⁡∣(x+1)x∣−1x+c\ln\left|\frac{\left(x+1\right)}{x}\right|-\frac{1}{x}+cln∣∣​x(x+1)​∣∣​−x1​+c﻿

Complete this statement: If f (x)=Cos x and g(x)=loglogx then ∫[f(x) -g(x)] dx = ___________.

﻿2x2cos⁡x−xlog⁡log⁡x+c2x^2\cos x-x\log\log x+c2x2cosx−xloglogx+c﻿

﻿sin⁡x+xlog⁡log⁡x−x+c\sin x+x\log\log x-x+csinx+xloglogx−x+c﻿

﻿cos⁡x−xlog⁡log⁡x+x+c\cos x-x\log\log x+x+ccosx−xloglogx+x+c﻿

Incorrect

Great!

Evaluate the integral $\int_{ }^{ }\frac{2x^2}{x+1}dx$

$x^2-2x\ln\left|4x\right|+c$

$4\ln\left|x\right|+c$

$x^2-2x+2\ln\left|x+1\right|+c$

Complete this statement:
If f (x)=2Cos x and g(x)=e^x then ∫[f(x) - g(x)] dx = ___________.

$\frac{x^2}{2}+e^x+c$

$2\sin x-e^x+c$

$2e^x+x+c$

Find the antiderivative of $\int_{ }^{ }\frac{1}{x^2+4x+8}dx$

$\frac{1}{2}\frac{x+2}{4}$

$\ln\left|x+1\right|+\frac{1}{x+1}+c$

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

Evaluate the integral $\int_{ }^{ }\frac{x+1}{x-2}dx$

$\left(x+3\right)\ln\left|x+1\right|+c$

$\left(x-3\right)\ln\left|x+1\right|+c$

$x+\ln\left|x-2\right|+c$

Complete this statement:
If f (x)=Cos x and g(x)=x³ then ∫[f(x) - g(x)] dx = ___________.

$\frac{x^2}{2}+\sin x+c$

$\sin x+\frac{1}{4}x^4+c$

$\sin x-\frac{1}{4}x^4+c$

Find the antiderivative of $\int_{ }^{ }\frac{x}{x^2+2x+1}dx$

$\ln\left|x+1\right|+c$

$\ln\left|x+1\right|+\frac{1}{x+1}+c$

$-\frac{1}{9x^9}+c$

Complete this statement:
If f (x)=e^x and g(x)=x³ then ∫[f(x) - g(x)] dx =___________.

$\frac{x^2}{2}+e^x+c$

$e^x-\frac{1}{4}x^4+c$

$e^x+\frac{1}{4}x^4+c$

Find the antiderivative of $\int_{ }^{ }\frac{2x}{x^2+2}dx$

$\ln\left|x^2+2\right|+\frac{1}{x}+c$

$\ln\left|x^2+2\right|+c$

$\ln\left|\frac{x^{ }+1}{x}\right|+c$

Find the antiderivative of $\int_{ }^{ }\frac{\left(1-2x\right)}{x^2\left(x+1\right)}dx$

$\ln\left|\frac{x+1\ }{x}\right|+\frac{1}{x}+c$

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

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

Complete this statement:
If f (x)=Cos x and g(x)=loglogx then ∫[f(x) -g(x)] dx = ___________.

$2x^2\cos x-x\log\log x+c$

$\sin x+x\log\log x-x+c$

$\cos x-x\log\log x+x+c$