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Integral Calculus

86785 \(\int x^{x} \log (e x) d x\) is equal to

1 \(x^{x}+c\)
2 \(\mathrm{x} \cdot \log \mathrm{x}+\mathrm{c}\)
3 \((\log \mathrm{x})^{\mathrm{x}}+\mathrm{c}\)
4 \(x^{\log x}+c\)
Manipal-2013
Integral Calculus

86786 Let \(f\) be a non-negative function in \([0,1]\) and twice differentiable in \((0,1)\). If
\(\int_{0}^{\mathrm{x}} \sqrt{1-\left(\mathrm{f}^{\prime}(\mathrm{t})\right)^{2}} \mathrm{dt}=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 1\)
and \(f(0)=0\), then \(\lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) d t\)

1 equals 0
2 equals 1
3 does not exist
4 equals \(1 / 2\)
JEE Main-2021-31.08.2021
Integral Calculus

86787 A function \(f\) is continuous for all \(x\) (and not everywhere zero) such \(f^{2}(x)=\int_{0}^{x} f(t) \frac{\operatorname{cost}}{2+\sin t} d t\) then \(f(x)\) is :

1 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\mathrm{x}+\cos \mathrm{x}}{2}\right) ; x \neq 0\)
2 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{3}{2+\cos \mathrm{x}}\right) ; \mathrm{x} \neq 0\)
3 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{2+\sin \mathrm{x}}{2}\right) ; \mathrm{x} \neq \mathrm{n} \pi, \mathrm{n} \in \mathrm{I}\)
4 \(\frac{\cos \mathrm{x}+\sin \mathrm{x}}{2+\sin \mathrm{x}} ; \mathrm{x} \neq \mathrm{n} \pi+\frac{3 \pi}{4}, \mathrm{n} \in \mathrm{I}\)
Assam CEE-2017
Integral Calculus

86788 If \(\int e^{2 x} f^{\prime}(x) d x=g(x)\), then
\(\int\left(e^{2 x} f(x)+e^{2 x} f^{\prime}(x)\right) d x=\)

1 \(\frac{1}{2}\left[e^{2 x} f(x)-g(x)\right]+C\)
2 \(\frac{1}{2}\left[e^{2 x} f(x)+g(x)\right]+C\)
3 \(\frac{1}{2}\left[\mathrm{e}^{2 \mathrm{x}} \mathrm{f}(2 \mathrm{x})+\mathrm{g}(\mathrm{x})\right]+\mathrm{C}\)
4 \(\frac{1}{2}\left[e^{2 x} f^{\prime}(2 x)+g(x)\right]+C\)
TS EAMCET-2017
NEET DIGITAL LIBRARY
Integral Calculus

86785 \(\int x^{x} \log (e x) d x\) is equal to

1 \(x^{x}+c\)
2 \(\mathrm{x} \cdot \log \mathrm{x}+\mathrm{c}\)
3 \((\log \mathrm{x})^{\mathrm{x}}+\mathrm{c}\)
4 \(x^{\log x}+c\)
Manipal-2013
Integral Calculus

86786 Let \(f\) be a non-negative function in \([0,1]\) and twice differentiable in \((0,1)\). If
\(\int_{0}^{\mathrm{x}} \sqrt{1-\left(\mathrm{f}^{\prime}(\mathrm{t})\right)^{2}} \mathrm{dt}=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 1\)
and \(f(0)=0\), then \(\lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) d t\)

1 equals 0
2 equals 1
3 does not exist
4 equals \(1 / 2\)
JEE Main-2021-31.08.2021
Integral Calculus

86787 A function \(f\) is continuous for all \(x\) (and not everywhere zero) such \(f^{2}(x)=\int_{0}^{x} f(t) \frac{\operatorname{cost}}{2+\sin t} d t\) then \(f(x)\) is :

1 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\mathrm{x}+\cos \mathrm{x}}{2}\right) ; x \neq 0\)
2 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{3}{2+\cos \mathrm{x}}\right) ; \mathrm{x} \neq 0\)
3 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{2+\sin \mathrm{x}}{2}\right) ; \mathrm{x} \neq \mathrm{n} \pi, \mathrm{n} \in \mathrm{I}\)
4 \(\frac{\cos \mathrm{x}+\sin \mathrm{x}}{2+\sin \mathrm{x}} ; \mathrm{x} \neq \mathrm{n} \pi+\frac{3 \pi}{4}, \mathrm{n} \in \mathrm{I}\)
Assam CEE-2017
Integral Calculus

86788 If \(\int e^{2 x} f^{\prime}(x) d x=g(x)\), then
\(\int\left(e^{2 x} f(x)+e^{2 x} f^{\prime}(x)\right) d x=\)

1 \(\frac{1}{2}\left[e^{2 x} f(x)-g(x)\right]+C\)
2 \(\frac{1}{2}\left[e^{2 x} f(x)+g(x)\right]+C\)
3 \(\frac{1}{2}\left[\mathrm{e}^{2 \mathrm{x}} \mathrm{f}(2 \mathrm{x})+\mathrm{g}(\mathrm{x})\right]+\mathrm{C}\)
4 \(\frac{1}{2}\left[e^{2 x} f^{\prime}(2 x)+g(x)\right]+C\)
TS EAMCET-2017
Join With Us for More Updates
Integral Calculus

86785 \(\int x^{x} \log (e x) d x\) is equal to

1 \(x^{x}+c\)
2 \(\mathrm{x} \cdot \log \mathrm{x}+\mathrm{c}\)
3 \((\log \mathrm{x})^{\mathrm{x}}+\mathrm{c}\)
4 \(x^{\log x}+c\)
Manipal-2013
Integral Calculus

86786 Let \(f\) be a non-negative function in \([0,1]\) and twice differentiable in \((0,1)\). If
\(\int_{0}^{\mathrm{x}} \sqrt{1-\left(\mathrm{f}^{\prime}(\mathrm{t})\right)^{2}} \mathrm{dt}=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 1\)
and \(f(0)=0\), then \(\lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) d t\)

1 equals 0
2 equals 1
3 does not exist
4 equals \(1 / 2\)
JEE Main-2021-31.08.2021
Integral Calculus

86787 A function \(f\) is continuous for all \(x\) (and not everywhere zero) such \(f^{2}(x)=\int_{0}^{x} f(t) \frac{\operatorname{cost}}{2+\sin t} d t\) then \(f(x)\) is :

1 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\mathrm{x}+\cos \mathrm{x}}{2}\right) ; x \neq 0\)
2 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{3}{2+\cos \mathrm{x}}\right) ; \mathrm{x} \neq 0\)
3 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{2+\sin \mathrm{x}}{2}\right) ; \mathrm{x} \neq \mathrm{n} \pi, \mathrm{n} \in \mathrm{I}\)
4 \(\frac{\cos \mathrm{x}+\sin \mathrm{x}}{2+\sin \mathrm{x}} ; \mathrm{x} \neq \mathrm{n} \pi+\frac{3 \pi}{4}, \mathrm{n} \in \mathrm{I}\)
Assam CEE-2017
Integral Calculus

86788 If \(\int e^{2 x} f^{\prime}(x) d x=g(x)\), then
\(\int\left(e^{2 x} f(x)+e^{2 x} f^{\prime}(x)\right) d x=\)

1 \(\frac{1}{2}\left[e^{2 x} f(x)-g(x)\right]+C\)
2 \(\frac{1}{2}\left[e^{2 x} f(x)+g(x)\right]+C\)
3 \(\frac{1}{2}\left[\mathrm{e}^{2 \mathrm{x}} \mathrm{f}(2 \mathrm{x})+\mathrm{g}(\mathrm{x})\right]+\mathrm{C}\)
4 \(\frac{1}{2}\left[e^{2 x} f^{\prime}(2 x)+g(x)\right]+C\)
TS EAMCET-2017
For More Updates join with Us
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Integral Calculus

86785 \(\int x^{x} \log (e x) d x\) is equal to

1 \(x^{x}+c\)
2 \(\mathrm{x} \cdot \log \mathrm{x}+\mathrm{c}\)
3 \((\log \mathrm{x})^{\mathrm{x}}+\mathrm{c}\)
4 \(x^{\log x}+c\)
Manipal-2013
Integral Calculus

86786 Let \(f\) be a non-negative function in \([0,1]\) and twice differentiable in \((0,1)\). If
\(\int_{0}^{\mathrm{x}} \sqrt{1-\left(\mathrm{f}^{\prime}(\mathrm{t})\right)^{2}} \mathrm{dt}=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 1\)
and \(f(0)=0\), then \(\lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) d t\)

1 equals 0
2 equals 1
3 does not exist
4 equals \(1 / 2\)
JEE Main-2021-31.08.2021
Integral Calculus

86787 A function \(f\) is continuous for all \(x\) (and not everywhere zero) such \(f^{2}(x)=\int_{0}^{x} f(t) \frac{\operatorname{cost}}{2+\sin t} d t\) then \(f(x)\) is :

1 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\mathrm{x}+\cos \mathrm{x}}{2}\right) ; x \neq 0\)
2 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{3}{2+\cos \mathrm{x}}\right) ; \mathrm{x} \neq 0\)
3 \(\frac{1}{2} \log _{\mathrm{e}}\left(\frac{2+\sin \mathrm{x}}{2}\right) ; \mathrm{x} \neq \mathrm{n} \pi, \mathrm{n} \in \mathrm{I}\)
4 \(\frac{\cos \mathrm{x}+\sin \mathrm{x}}{2+\sin \mathrm{x}} ; \mathrm{x} \neq \mathrm{n} \pi+\frac{3 \pi}{4}, \mathrm{n} \in \mathrm{I}\)
Assam CEE-2017
Integral Calculus

86788 If \(\int e^{2 x} f^{\prime}(x) d x=g(x)\), then
\(\int\left(e^{2 x} f(x)+e^{2 x} f^{\prime}(x)\right) d x=\)

1 \(\frac{1}{2}\left[e^{2 x} f(x)-g(x)\right]+C\)
2 \(\frac{1}{2}\left[e^{2 x} f(x)+g(x)\right]+C\)
3 \(\frac{1}{2}\left[\mathrm{e}^{2 \mathrm{x}} \mathrm{f}(2 \mathrm{x})+\mathrm{g}(\mathrm{x})\right]+\mathrm{C}\)
4 \(\frac{1}{2}\left[e^{2 x} f^{\prime}(2 x)+g(x)\right]+C\)
TS EAMCET-2017