Theorem of Definite Integrals and its Properties
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Integral Calculus

86468 The value of \(\int \frac{e^{x}\left(x^{2} \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^{2}+1} d x\) is equal to

1 \(\mathrm{e}^{\mathrm{x}} \tan ^{-1} \mathrm{x}+\mathrm{c}\)
2 \(\tan ^{-1}\left(e^{x}\right)+c\)
3 \(\tan ^{-1}\left(x^{e}\right)+c\)
4 \(\mathrm{e}^{\tan ^{-1} \mathrm{x}}+\mathrm{c}\)
Karnataka CET-2016
Integral Calculus

86469 \(\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x\) is equal to

1 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x^{2}}+C\)
2 \(\frac{-\left(1+x^{4}\right)^{3 / 4}}{x}+C\)
3 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x}+C\)
4 \(\frac{-\left(1+\mathrm{x}^{4}\right)^{1 / 4}}{2 \mathrm{x}}+\mathrm{C}\)
Karnataka CET-2015
Integral Calculus

86470 If \(f(x)=f(\pi+e-x)\) and \(\int_{e}^{\pi} f(x) d x=\frac{2}{e+\pi}\), then \(\int_{e}^{\pi} x f(x) d x\) is equal to

1 \(\pi-\mathrm{e}\)
2 \(\frac{\pi+\mathrm{e}}{2}\)
3 1
4 \(\frac{\pi-\mathrm{e}}{2}\)
Karnataka CET-2014
Integral Calculus

86471 When \(x>0\), then \(\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x\) is

1 \(2\left[x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
2 \(2\left[x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
3 \(2 x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
4 \(2 x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
Karnataka CET-2011
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Integral Calculus

86468 The value of \(\int \frac{e^{x}\left(x^{2} \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^{2}+1} d x\) is equal to

1 \(\mathrm{e}^{\mathrm{x}} \tan ^{-1} \mathrm{x}+\mathrm{c}\)
2 \(\tan ^{-1}\left(e^{x}\right)+c\)
3 \(\tan ^{-1}\left(x^{e}\right)+c\)
4 \(\mathrm{e}^{\tan ^{-1} \mathrm{x}}+\mathrm{c}\)
Karnataka CET-2016
Integral Calculus

86469 \(\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x\) is equal to

1 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x^{2}}+C\)
2 \(\frac{-\left(1+x^{4}\right)^{3 / 4}}{x}+C\)
3 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x}+C\)
4 \(\frac{-\left(1+\mathrm{x}^{4}\right)^{1 / 4}}{2 \mathrm{x}}+\mathrm{C}\)
Karnataka CET-2015
Integral Calculus

86470 If \(f(x)=f(\pi+e-x)\) and \(\int_{e}^{\pi} f(x) d x=\frac{2}{e+\pi}\), then \(\int_{e}^{\pi} x f(x) d x\) is equal to

1 \(\pi-\mathrm{e}\)
2 \(\frac{\pi+\mathrm{e}}{2}\)
3 1
4 \(\frac{\pi-\mathrm{e}}{2}\)
Karnataka CET-2014
Integral Calculus

86471 When \(x>0\), then \(\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x\) is

1 \(2\left[x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
2 \(2\left[x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
3 \(2 x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
4 \(2 x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
Karnataka CET-2011
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Integral Calculus

86468 The value of \(\int \frac{e^{x}\left(x^{2} \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^{2}+1} d x\) is equal to

1 \(\mathrm{e}^{\mathrm{x}} \tan ^{-1} \mathrm{x}+\mathrm{c}\)
2 \(\tan ^{-1}\left(e^{x}\right)+c\)
3 \(\tan ^{-1}\left(x^{e}\right)+c\)
4 \(\mathrm{e}^{\tan ^{-1} \mathrm{x}}+\mathrm{c}\)
Karnataka CET-2016
Integral Calculus

86469 \(\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x\) is equal to

1 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x^{2}}+C\)
2 \(\frac{-\left(1+x^{4}\right)^{3 / 4}}{x}+C\)
3 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x}+C\)
4 \(\frac{-\left(1+\mathrm{x}^{4}\right)^{1 / 4}}{2 \mathrm{x}}+\mathrm{C}\)
Karnataka CET-2015
Integral Calculus

86470 If \(f(x)=f(\pi+e-x)\) and \(\int_{e}^{\pi} f(x) d x=\frac{2}{e+\pi}\), then \(\int_{e}^{\pi} x f(x) d x\) is equal to

1 \(\pi-\mathrm{e}\)
2 \(\frac{\pi+\mathrm{e}}{2}\)
3 1
4 \(\frac{\pi-\mathrm{e}}{2}\)
Karnataka CET-2014
Integral Calculus

86471 When \(x>0\), then \(\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x\) is

1 \(2\left[x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
2 \(2\left[x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
3 \(2 x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
4 \(2 x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
Karnataka CET-2011
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Integral Calculus

86468 The value of \(\int \frac{e^{x}\left(x^{2} \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^{2}+1} d x\) is equal to

1 \(\mathrm{e}^{\mathrm{x}} \tan ^{-1} \mathrm{x}+\mathrm{c}\)
2 \(\tan ^{-1}\left(e^{x}\right)+c\)
3 \(\tan ^{-1}\left(x^{e}\right)+c\)
4 \(\mathrm{e}^{\tan ^{-1} \mathrm{x}}+\mathrm{c}\)
Karnataka CET-2016
Integral Calculus

86469 \(\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x\) is equal to

1 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x^{2}}+C\)
2 \(\frac{-\left(1+x^{4}\right)^{3 / 4}}{x}+C\)
3 \(\frac{-\left(1+x^{4}\right)^{1 / 4}}{x}+C\)
4 \(\frac{-\left(1+\mathrm{x}^{4}\right)^{1 / 4}}{2 \mathrm{x}}+\mathrm{C}\)
Karnataka CET-2015
Integral Calculus

86470 If \(f(x)=f(\pi+e-x)\) and \(\int_{e}^{\pi} f(x) d x=\frac{2}{e+\pi}\), then \(\int_{e}^{\pi} x f(x) d x\) is equal to

1 \(\pi-\mathrm{e}\)
2 \(\frac{\pi+\mathrm{e}}{2}\)
3 1
4 \(\frac{\pi-\mathrm{e}}{2}\)
Karnataka CET-2014
Integral Calculus

86471 When \(x>0\), then \(\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x\) is

1 \(2\left[x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
2 \(2\left[x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)\right]+\mathrm{c}\)
3 \(2 x \tan ^{-1} \mathrm{x}+\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
4 \(2 x \tan ^{-1} \mathrm{x}-\log \left(1+\mathrm{x}^{2}\right)+\mathrm{c}\)
Karnataka CET-2011