Integration by Parts
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NEET DIGITAL LIBRARY
Integral Calculus

86287 \(\int \frac{x^{3}-1}{x^{3}+x} d x=\)

1 \(x+\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)+\sin ^{-1}(x)+c\)
2 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\sin ^{-1}(x)+c\)
3 \(\mathrm{x}+\log |\mathrm{x}|-\frac{1}{2} \log \left(\mathrm{x}^{2}+1\right)+\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
4 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
AP EAMCET-2020-17.09.2020
Integral Calculus

86288 \(\int \frac{x-1}{(x+1) \sqrt{x^{3}+x^{2}+x}} d x=\)

1 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
2 \(\tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
3 \(\tan ^{-1}\left(\sqrt{\frac{x}{1+x+x^{2}}}\right)+c\)
4 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86289 If \(I(x)=\int x^{2}(\log x)^{2} d x\) and \(I(1)=0\), then \(I(x)\)

1 \(\frac{x^{3}}{18}\left[8(\log x)^{2}-3 \log x\right]+\frac{7}{18}\)
2 \(\frac{x^{3}}{27}\left[9(\log x)^{2}+6 \log x\right]-\frac{2}{27}\)
3 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x+2\right]-\frac{2}{27}\)
4 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x-2\right]+\frac{2}{27}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86290 \(\int \frac{x^{5} d x}{\left(x^{2}+x+1\right)\left(x^{6}+1\right)\left(x^{4}-x^{3}+x-1\right)}=\)

1 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
2 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
3 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{4}+1}{\mathrm{x}^{4}-1}\right|+\mathrm{c}\)
4 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{8}+4}{\mathrm{x}^{6}-1}\right|+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
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Integral Calculus

86287 \(\int \frac{x^{3}-1}{x^{3}+x} d x=\)

1 \(x+\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)+\sin ^{-1}(x)+c\)
2 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\sin ^{-1}(x)+c\)
3 \(\mathrm{x}+\log |\mathrm{x}|-\frac{1}{2} \log \left(\mathrm{x}^{2}+1\right)+\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
4 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
AP EAMCET-2020-17.09.2020
Integral Calculus

86288 \(\int \frac{x-1}{(x+1) \sqrt{x^{3}+x^{2}+x}} d x=\)

1 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
2 \(\tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
3 \(\tan ^{-1}\left(\sqrt{\frac{x}{1+x+x^{2}}}\right)+c\)
4 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86289 If \(I(x)=\int x^{2}(\log x)^{2} d x\) and \(I(1)=0\), then \(I(x)\)

1 \(\frac{x^{3}}{18}\left[8(\log x)^{2}-3 \log x\right]+\frac{7}{18}\)
2 \(\frac{x^{3}}{27}\left[9(\log x)^{2}+6 \log x\right]-\frac{2}{27}\)
3 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x+2\right]-\frac{2}{27}\)
4 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x-2\right]+\frac{2}{27}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86290 \(\int \frac{x^{5} d x}{\left(x^{2}+x+1\right)\left(x^{6}+1\right)\left(x^{4}-x^{3}+x-1\right)}=\)

1 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
2 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
3 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{4}+1}{\mathrm{x}^{4}-1}\right|+\mathrm{c}\)
4 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{8}+4}{\mathrm{x}^{6}-1}\right|+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
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Integral Calculus

86287 \(\int \frac{x^{3}-1}{x^{3}+x} d x=\)

1 \(x+\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)+\sin ^{-1}(x)+c\)
2 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\sin ^{-1}(x)+c\)
3 \(\mathrm{x}+\log |\mathrm{x}|-\frac{1}{2} \log \left(\mathrm{x}^{2}+1\right)+\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
4 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
AP EAMCET-2020-17.09.2020
Integral Calculus

86288 \(\int \frac{x-1}{(x+1) \sqrt{x^{3}+x^{2}+x}} d x=\)

1 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
2 \(\tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
3 \(\tan ^{-1}\left(\sqrt{\frac{x}{1+x+x^{2}}}\right)+c\)
4 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86289 If \(I(x)=\int x^{2}(\log x)^{2} d x\) and \(I(1)=0\), then \(I(x)\)

1 \(\frac{x^{3}}{18}\left[8(\log x)^{2}-3 \log x\right]+\frac{7}{18}\)
2 \(\frac{x^{3}}{27}\left[9(\log x)^{2}+6 \log x\right]-\frac{2}{27}\)
3 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x+2\right]-\frac{2}{27}\)
4 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x-2\right]+\frac{2}{27}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86290 \(\int \frac{x^{5} d x}{\left(x^{2}+x+1\right)\left(x^{6}+1\right)\left(x^{4}-x^{3}+x-1\right)}=\)

1 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
2 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
3 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{4}+1}{\mathrm{x}^{4}-1}\right|+\mathrm{c}\)
4 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{8}+4}{\mathrm{x}^{6}-1}\right|+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
NEET DIGITAL LIBRARY
Integral Calculus

86287 \(\int \frac{x^{3}-1}{x^{3}+x} d x=\)

1 \(x+\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)+\sin ^{-1}(x)+c\)
2 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\sin ^{-1}(x)+c\)
3 \(\mathrm{x}+\log |\mathrm{x}|-\frac{1}{2} \log \left(\mathrm{x}^{2}+1\right)+\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
4 \(x-\log |x|+\frac{1}{2} \log \left(x^{2}+1\right)-\tan ^{-1}(\mathrm{x})+\mathrm{c}\)
AP EAMCET-2020-17.09.2020
Integral Calculus

86288 \(\int \frac{x-1}{(x+1) \sqrt{x^{3}+x^{2}+x}} d x=\)

1 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
2 \(\tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
3 \(\tan ^{-1}\left(\sqrt{\frac{x}{1+x+x^{2}}}\right)+c\)
4 \(2 \tan ^{-1}\left(\sqrt{\frac{1+\mathrm{x}^{2}}{\mathrm{x}}}\right)+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86289 If \(I(x)=\int x^{2}(\log x)^{2} d x\) and \(I(1)=0\), then \(I(x)\)

1 \(\frac{x^{3}}{18}\left[8(\log x)^{2}-3 \log x\right]+\frac{7}{18}\)
2 \(\frac{x^{3}}{27}\left[9(\log x)^{2}+6 \log x\right]-\frac{2}{27}\)
3 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x+2\right]-\frac{2}{27}\)
4 \(\frac{x^{3}}{27}\left[9(\log x)^{2}-6 \log x-2\right]+\frac{2}{27}\)
AP EAMCET-2019-20.04.2019
Integral Calculus

86290 \(\int \frac{x^{5} d x}{\left(x^{2}+x+1\right)\left(x^{6}+1\right)\left(x^{4}-x^{3}+x-1\right)}=\)

1 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
2 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{6}-1}{\mathrm{x}^{6}+1}\right|+\mathrm{c}\)
3 \(\frac{1}{12} \log _{\mathrm{e}}\left|\frac{\mathrm{x}^{4}+1}{\mathrm{x}^{4}-1}\right|+\mathrm{c}\)
4 \(\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{8}+4}{\mathrm{x}^{6}-1}\right|+\mathrm{c}\)
AP EAMCET-2019-20.04.2019
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