Theorem of Definite Integrals and its Properties
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NEET DIGITAL LIBRARY
Integral Calculus

86552 The integral \(\int_{-1}^{3}\left(\tan ^{-1} \frac{x}{x^{2}+1}+\tan ^{-1} \frac{x^{2}+1}{x}\right) d x\) is equal to

1 \(\pi / 4\)
2 \(\pi / 2\)
3 \(\pi\)
4 \(2 \pi\)
AMU-2005
Integral Calculus

86553 The value of the integral \(\int_{0}^{2 a} \frac{f(x)}{f(x)+f(2 a-x)} d x\) is

1 0
2 a
3 \(2 \mathrm{a}\)
4 None of these
AMU-2005
Integral Calculus

86554 If \(\phi(a-x)=\phi(x)\), then what is \(\int_{0}^{a} x \phi(x) d x\) equal to?

1 \(a \int_{0}^{a} \phi(x) d x\)
2 \(x \int_{0}^{a} \phi(x) d x\)
3 \(2 \mathrm{a} \int_{0}^{\frac{\mathrm{a}}{2}} \phi(\mathrm{x}) \mathrm{dx}\)
4 \(\frac{a}{2} \int_{0}^{a} \phi(x) d x\)
SCRA-2009
Integral Calculus

86557 What is \(\int_{0}^{1} \frac{\mathrm{x}^{2} \mathrm{dx}}{\sqrt{\mathrm{x}^{6}+1}}\) equal to?

1 \(\frac{1}{3} \ln (\sqrt{2}+1)\)
2 \(\frac{1}{2} \ln (\sqrt{2}+1)\)
3 \(\frac{1}{3} \ln (\sqrt{2}-1)\)
4 \(\frac{1}{2} \ln (\sqrt{2}-1)\)
SCRA-2012
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Integral Calculus

86552 The integral \(\int_{-1}^{3}\left(\tan ^{-1} \frac{x}{x^{2}+1}+\tan ^{-1} \frac{x^{2}+1}{x}\right) d x\) is equal to

1 \(\pi / 4\)
2 \(\pi / 2\)
3 \(\pi\)
4 \(2 \pi\)
AMU-2005
Integral Calculus

86553 The value of the integral \(\int_{0}^{2 a} \frac{f(x)}{f(x)+f(2 a-x)} d x\) is

1 0
2 a
3 \(2 \mathrm{a}\)
4 None of these
AMU-2005
Integral Calculus

86554 If \(\phi(a-x)=\phi(x)\), then what is \(\int_{0}^{a} x \phi(x) d x\) equal to?

1 \(a \int_{0}^{a} \phi(x) d x\)
2 \(x \int_{0}^{a} \phi(x) d x\)
3 \(2 \mathrm{a} \int_{0}^{\frac{\mathrm{a}}{2}} \phi(\mathrm{x}) \mathrm{dx}\)
4 \(\frac{a}{2} \int_{0}^{a} \phi(x) d x\)
SCRA-2009
Integral Calculus

86557 What is \(\int_{0}^{1} \frac{\mathrm{x}^{2} \mathrm{dx}}{\sqrt{\mathrm{x}^{6}+1}}\) equal to?

1 \(\frac{1}{3} \ln (\sqrt{2}+1)\)
2 \(\frac{1}{2} \ln (\sqrt{2}+1)\)
3 \(\frac{1}{3} \ln (\sqrt{2}-1)\)
4 \(\frac{1}{2} \ln (\sqrt{2}-1)\)
SCRA-2012
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Integral Calculus

86552 The integral \(\int_{-1}^{3}\left(\tan ^{-1} \frac{x}{x^{2}+1}+\tan ^{-1} \frac{x^{2}+1}{x}\right) d x\) is equal to

1 \(\pi / 4\)
2 \(\pi / 2\)
3 \(\pi\)
4 \(2 \pi\)
AMU-2005
Integral Calculus

86553 The value of the integral \(\int_{0}^{2 a} \frac{f(x)}{f(x)+f(2 a-x)} d x\) is

1 0
2 a
3 \(2 \mathrm{a}\)
4 None of these
AMU-2005
Integral Calculus

86554 If \(\phi(a-x)=\phi(x)\), then what is \(\int_{0}^{a} x \phi(x) d x\) equal to?

1 \(a \int_{0}^{a} \phi(x) d x\)
2 \(x \int_{0}^{a} \phi(x) d x\)
3 \(2 \mathrm{a} \int_{0}^{\frac{\mathrm{a}}{2}} \phi(\mathrm{x}) \mathrm{dx}\)
4 \(\frac{a}{2} \int_{0}^{a} \phi(x) d x\)
SCRA-2009
Integral Calculus

86557 What is \(\int_{0}^{1} \frac{\mathrm{x}^{2} \mathrm{dx}}{\sqrt{\mathrm{x}^{6}+1}}\) equal to?

1 \(\frac{1}{3} \ln (\sqrt{2}+1)\)
2 \(\frac{1}{2} \ln (\sqrt{2}+1)\)
3 \(\frac{1}{3} \ln (\sqrt{2}-1)\)
4 \(\frac{1}{2} \ln (\sqrt{2}-1)\)
SCRA-2012
NEET DIGITAL LIBRARY
Integral Calculus

86552 The integral \(\int_{-1}^{3}\left(\tan ^{-1} \frac{x}{x^{2}+1}+\tan ^{-1} \frac{x^{2}+1}{x}\right) d x\) is equal to

1 \(\pi / 4\)
2 \(\pi / 2\)
3 \(\pi\)
4 \(2 \pi\)
AMU-2005
Integral Calculus

86553 The value of the integral \(\int_{0}^{2 a} \frac{f(x)}{f(x)+f(2 a-x)} d x\) is

1 0
2 a
3 \(2 \mathrm{a}\)
4 None of these
AMU-2005
Integral Calculus

86554 If \(\phi(a-x)=\phi(x)\), then what is \(\int_{0}^{a} x \phi(x) d x\) equal to?

1 \(a \int_{0}^{a} \phi(x) d x\)
2 \(x \int_{0}^{a} \phi(x) d x\)
3 \(2 \mathrm{a} \int_{0}^{\frac{\mathrm{a}}{2}} \phi(\mathrm{x}) \mathrm{dx}\)
4 \(\frac{a}{2} \int_{0}^{a} \phi(x) d x\)
SCRA-2009
Integral Calculus

86557 What is \(\int_{0}^{1} \frac{\mathrm{x}^{2} \mathrm{dx}}{\sqrt{\mathrm{x}^{6}+1}}\) equal to?

1 \(\frac{1}{3} \ln (\sqrt{2}+1)\)
2 \(\frac{1}{2} \ln (\sqrt{2}+1)\)
3 \(\frac{1}{3} \ln (\sqrt{2}-1)\)
4 \(\frac{1}{2} \ln (\sqrt{2}-1)\)
SCRA-2012
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