Integration by Parts
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Integral Calculus

86254 Evaluate: \(\int \frac{1}{\sin x+\sqrt{3} \cos x} d x\)

1 \(\log \left|\tan \left(\frac{\mathrm{x}}{2}\right)\right|+\mathrm{C}\)
2 \(-\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
3 \(\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
4 None of these
COMEDK-2017
Integral Calculus

86302 \(\int \frac{x^{3} d x}{1+x^{4}}\) equals

1 \(\log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
2 \(\frac{1}{4} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
3 \(\frac{1}{2} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
4 None of the above
AMU-2016
Integral Calculus

86303 The value of the integral \(\int_{0}^{1} \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{\log _{e} x^{3}}-e^{\log _{e} x^{2}}} d x\) is

1 \(1 / 3\)
2 1
3 \(-1 / 3\)
4 -1
AMU-2016
Integral Calculus

86361 \(\int e^{x \operatorname{loga}} e^{x} d x\) is equal to :

1 \(\frac{\mathrm{a}^{\mathrm{x}}}{\log \mathrm{ae}}+\mathrm{c}\)
2 \(\frac{\mathrm{e}^{\mathrm{x}}}{1+\log _{\mathrm{e}} \mathrm{a}}+\mathrm{c}\)
3 \((a)^{\mathrm{x}}+\mathrm{c}\)
4 \(\frac{(\mathrm{ae})^{\mathrm{x}}}{\log _{\mathrm{e}} \mathrm{ae}}+\mathrm{c}\)
5 \(\frac{a^{x} e^{x}}{\log _{x} a}+c\)
Kerala CEE-2005
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Integral Calculus

86254 Evaluate: \(\int \frac{1}{\sin x+\sqrt{3} \cos x} d x\)

1 \(\log \left|\tan \left(\frac{\mathrm{x}}{2}\right)\right|+\mathrm{C}\)
2 \(-\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
3 \(\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
4 None of these
COMEDK-2017
Integral Calculus

86302 \(\int \frac{x^{3} d x}{1+x^{4}}\) equals

1 \(\log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
2 \(\frac{1}{4} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
3 \(\frac{1}{2} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
4 None of the above
AMU-2016
Integral Calculus

86303 The value of the integral \(\int_{0}^{1} \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{\log _{e} x^{3}}-e^{\log _{e} x^{2}}} d x\) is

1 \(1 / 3\)
2 1
3 \(-1 / 3\)
4 -1
AMU-2016
Integral Calculus

86361 \(\int e^{x \operatorname{loga}} e^{x} d x\) is equal to :

1 \(\frac{\mathrm{a}^{\mathrm{x}}}{\log \mathrm{ae}}+\mathrm{c}\)
2 \(\frac{\mathrm{e}^{\mathrm{x}}}{1+\log _{\mathrm{e}} \mathrm{a}}+\mathrm{c}\)
3 \((a)^{\mathrm{x}}+\mathrm{c}\)
4 \(\frac{(\mathrm{ae})^{\mathrm{x}}}{\log _{\mathrm{e}} \mathrm{ae}}+\mathrm{c}\)
5 \(\frac{a^{x} e^{x}}{\log _{x} a}+c\)
Kerala CEE-2005
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Integral Calculus

86254 Evaluate: \(\int \frac{1}{\sin x+\sqrt{3} \cos x} d x\)

1 \(\log \left|\tan \left(\frac{\mathrm{x}}{2}\right)\right|+\mathrm{C}\)
2 \(-\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
3 \(\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
4 None of these
COMEDK-2017
Integral Calculus

86302 \(\int \frac{x^{3} d x}{1+x^{4}}\) equals

1 \(\log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
2 \(\frac{1}{4} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
3 \(\frac{1}{2} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
4 None of the above
AMU-2016
Integral Calculus

86303 The value of the integral \(\int_{0}^{1} \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{\log _{e} x^{3}}-e^{\log _{e} x^{2}}} d x\) is

1 \(1 / 3\)
2 1
3 \(-1 / 3\)
4 -1
AMU-2016
Integral Calculus

86361 \(\int e^{x \operatorname{loga}} e^{x} d x\) is equal to :

1 \(\frac{\mathrm{a}^{\mathrm{x}}}{\log \mathrm{ae}}+\mathrm{c}\)
2 \(\frac{\mathrm{e}^{\mathrm{x}}}{1+\log _{\mathrm{e}} \mathrm{a}}+\mathrm{c}\)
3 \((a)^{\mathrm{x}}+\mathrm{c}\)
4 \(\frac{(\mathrm{ae})^{\mathrm{x}}}{\log _{\mathrm{e}} \mathrm{ae}}+\mathrm{c}\)
5 \(\frac{a^{x} e^{x}}{\log _{x} a}+c\)
Kerala CEE-2005
NEET DIGITAL LIBRARY
Integral Calculus

86254 Evaluate: \(\int \frac{1}{\sin x+\sqrt{3} \cos x} d x\)

1 \(\log \left|\tan \left(\frac{\mathrm{x}}{2}\right)\right|+\mathrm{C}\)
2 \(-\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
3 \(\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{C}\)
4 None of these
COMEDK-2017
Integral Calculus

86302 \(\int \frac{x^{3} d x}{1+x^{4}}\) equals

1 \(\log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
2 \(\frac{1}{4} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
3 \(\frac{1}{2} \log \left(\mathrm{x}^{4}+1\right)+\mathrm{C}\)
4 None of the above
AMU-2016
Integral Calculus

86303 The value of the integral \(\int_{0}^{1} \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{\log _{e} x^{3}}-e^{\log _{e} x^{2}}} d x\) is

1 \(1 / 3\)
2 1
3 \(-1 / 3\)
4 -1
AMU-2016
Integral Calculus

86361 \(\int e^{x \operatorname{loga}} e^{x} d x\) is equal to :

1 \(\frac{\mathrm{a}^{\mathrm{x}}}{\log \mathrm{ae}}+\mathrm{c}\)
2 \(\frac{\mathrm{e}^{\mathrm{x}}}{1+\log _{\mathrm{e}} \mathrm{a}}+\mathrm{c}\)
3 \((a)^{\mathrm{x}}+\mathrm{c}\)
4 \(\frac{(\mathrm{ae})^{\mathrm{x}}}{\log _{\mathrm{e}} \mathrm{ae}}+\mathrm{c}\)
5 \(\frac{a^{x} e^{x}}{\log _{x} a}+c\)
Kerala CEE-2005
Join With Us for More Updates