Definite Integral as Limit of a Sum
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Integral Calculus

86382 \(\int_{0}^{1} \frac{\mathrm{dx}}{\sqrt{1+\mathrm{x}}+\sqrt{\mathrm{x}}}\) is equal to

1 \(\frac{4}{3}(\sqrt{2}-1)\)
2 \(\frac{3}{4}(\sqrt{2}-1)\)
3 \(\frac{4}{3}(1-\sqrt{2})\)
4 \(\frac{3}{4}(1-\sqrt{2})\)
UPSEE-2010
Integral Calculus

86384 The value of \(\int_{0}^{1} \frac{x^{4}+1}{x^{2}+1} d x\) is

1 \(\frac{1}{6}(3-4 \pi)\)
2 \(\frac{1}{6}(3 \pi+4)\)
3 \(\frac{1}{6}(3+4 \pi)\)
4 \(\frac{1}{6}(3 \pi-4)\)
UPSEE-2007
Integral Calculus

86385 The value of \(\int_{2}^{3} \frac{x+1}{x^{2}(x-1)} d x\) is

1 \(\log \frac{16}{9}+\frac{1}{6}\)
2 \(\log \frac{16}{9}-\frac{1}{6}\)
3 \(2 \log 2-\frac{1}{6}\)
4 \(\log \frac{4}{3}-\frac{1}{6}\)
UPSEE-2007
Integral Calculus

86386 If \(I_{m, n}=\int_{0}^{1} x^{m}(\log x)^{n} d x\), then it is equal to

1 \(\frac{n}{n+1} I_{m, n-1}\)
2 \(\frac{-m}{n+1} I_{m, n-1}\)
3 \(\frac{-n}{m+1} I_{m, n-1}\)
4 \(\frac{\mathrm{m}}{\mathrm{n}+1} \mathrm{I}_{\mathrm{m}, \mathrm{n}-1}\)
BCECE-2011
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Integral Calculus

86382 \(\int_{0}^{1} \frac{\mathrm{dx}}{\sqrt{1+\mathrm{x}}+\sqrt{\mathrm{x}}}\) is equal to

1 \(\frac{4}{3}(\sqrt{2}-1)\)
2 \(\frac{3}{4}(\sqrt{2}-1)\)
3 \(\frac{4}{3}(1-\sqrt{2})\)
4 \(\frac{3}{4}(1-\sqrt{2})\)
UPSEE-2010
Integral Calculus

86384 The value of \(\int_{0}^{1} \frac{x^{4}+1}{x^{2}+1} d x\) is

1 \(\frac{1}{6}(3-4 \pi)\)
2 \(\frac{1}{6}(3 \pi+4)\)
3 \(\frac{1}{6}(3+4 \pi)\)
4 \(\frac{1}{6}(3 \pi-4)\)
UPSEE-2007
Integral Calculus

86385 The value of \(\int_{2}^{3} \frac{x+1}{x^{2}(x-1)} d x\) is

1 \(\log \frac{16}{9}+\frac{1}{6}\)
2 \(\log \frac{16}{9}-\frac{1}{6}\)
3 \(2 \log 2-\frac{1}{6}\)
4 \(\log \frac{4}{3}-\frac{1}{6}\)
UPSEE-2007
Integral Calculus

86386 If \(I_{m, n}=\int_{0}^{1} x^{m}(\log x)^{n} d x\), then it is equal to

1 \(\frac{n}{n+1} I_{m, n-1}\)
2 \(\frac{-m}{n+1} I_{m, n-1}\)
3 \(\frac{-n}{m+1} I_{m, n-1}\)
4 \(\frac{\mathrm{m}}{\mathrm{n}+1} \mathrm{I}_{\mathrm{m}, \mathrm{n}-1}\)
BCECE-2011
NEET DIGITAL LIBRARY
Integral Calculus

86382 \(\int_{0}^{1} \frac{\mathrm{dx}}{\sqrt{1+\mathrm{x}}+\sqrt{\mathrm{x}}}\) is equal to

1 \(\frac{4}{3}(\sqrt{2}-1)\)
2 \(\frac{3}{4}(\sqrt{2}-1)\)
3 \(\frac{4}{3}(1-\sqrt{2})\)
4 \(\frac{3}{4}(1-\sqrt{2})\)
UPSEE-2010
Integral Calculus

86384 The value of \(\int_{0}^{1} \frac{x^{4}+1}{x^{2}+1} d x\) is

1 \(\frac{1}{6}(3-4 \pi)\)
2 \(\frac{1}{6}(3 \pi+4)\)
3 \(\frac{1}{6}(3+4 \pi)\)
4 \(\frac{1}{6}(3 \pi-4)\)
UPSEE-2007
Integral Calculus

86385 The value of \(\int_{2}^{3} \frac{x+1}{x^{2}(x-1)} d x\) is

1 \(\log \frac{16}{9}+\frac{1}{6}\)
2 \(\log \frac{16}{9}-\frac{1}{6}\)
3 \(2 \log 2-\frac{1}{6}\)
4 \(\log \frac{4}{3}-\frac{1}{6}\)
UPSEE-2007
Integral Calculus

86386 If \(I_{m, n}=\int_{0}^{1} x^{m}(\log x)^{n} d x\), then it is equal to

1 \(\frac{n}{n+1} I_{m, n-1}\)
2 \(\frac{-m}{n+1} I_{m, n-1}\)
3 \(\frac{-n}{m+1} I_{m, n-1}\)
4 \(\frac{\mathrm{m}}{\mathrm{n}+1} \mathrm{I}_{\mathrm{m}, \mathrm{n}-1}\)
BCECE-2011
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Integral Calculus

86382 \(\int_{0}^{1} \frac{\mathrm{dx}}{\sqrt{1+\mathrm{x}}+\sqrt{\mathrm{x}}}\) is equal to

1 \(\frac{4}{3}(\sqrt{2}-1)\)
2 \(\frac{3}{4}(\sqrt{2}-1)\)
3 \(\frac{4}{3}(1-\sqrt{2})\)
4 \(\frac{3}{4}(1-\sqrt{2})\)
UPSEE-2010
Integral Calculus

86384 The value of \(\int_{0}^{1} \frac{x^{4}+1}{x^{2}+1} d x\) is

1 \(\frac{1}{6}(3-4 \pi)\)
2 \(\frac{1}{6}(3 \pi+4)\)
3 \(\frac{1}{6}(3+4 \pi)\)
4 \(\frac{1}{6}(3 \pi-4)\)
UPSEE-2007
Integral Calculus

86385 The value of \(\int_{2}^{3} \frac{x+1}{x^{2}(x-1)} d x\) is

1 \(\log \frac{16}{9}+\frac{1}{6}\)
2 \(\log \frac{16}{9}-\frac{1}{6}\)
3 \(2 \log 2-\frac{1}{6}\)
4 \(\log \frac{4}{3}-\frac{1}{6}\)
UPSEE-2007
Integral Calculus

86386 If \(I_{m, n}=\int_{0}^{1} x^{m}(\log x)^{n} d x\), then it is equal to

1 \(\frac{n}{n+1} I_{m, n-1}\)
2 \(\frac{-m}{n+1} I_{m, n-1}\)
3 \(\frac{-n}{m+1} I_{m, n-1}\)
4 \(\frac{\mathrm{m}}{\mathrm{n}+1} \mathrm{I}_{\mathrm{m}, \mathrm{n}-1}\)
BCECE-2011
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