86372
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x=\)
1 \(\frac{\pi}{4}\)
2 \(\frac{3 \pi}{4}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{3}\)
Explanation:
(C) : \(I=\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \sin ^{2}(-x) d x \Rightarrow I=2 \int_{0}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \frac{1-\cos 2 x}{2} d x \Rightarrow I=\int_{0}^{\pi / 2} 1-\cos 2 x d x\) \(I=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\frac{\pi}{2}\)
MHT CET-2020
Integral Calculus
86383
\(\int_{0}^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}\) is
1 \(\mathrm{e}^{1000}-1\)
2 \(\frac{e^{1000}-1}{e-1}\)
3 \(1000(\mathrm{e}-1)\)
4 \(\frac{\mathrm{e}-1}{1000}\)
Explanation:
(C) : Given, \(\int_{0}^{1000} e^{x-[x]} d x=1000 \int_{0}^{1} e^{x-[x]} d x\) \([\) Step function \((\mathrm{x})]=0\) for \((0,1)=1\) for \((1,2)\) and so on \(\}\) \(=1000 \int_{0}^{1} \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}=1000\left[\mathrm{e}^{\mathrm{x}}\right]_{0}^{1}=1000\left[\mathrm{e}^{1}-\mathrm{e}^{0}\right]\) \(=1000(\mathrm{e}-1)\)
UPSEE-2008
Integral Calculus
86401
\(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x=\)
1 \(\frac{5}{3}\)
2 \(\frac{3}{5}\)
3 1
4 -1
Explanation:
(C) : \(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x\) \(=\int_{1}^{2} \frac{x^{3}}{x^{2}} d x-\int_{1}^{2} \frac{1}{x^{2}} d x =\int_{1}^{2} x d x-\int_{1}^{2} x^{-2} d x\) \(=\left[\frac{x^{2}}{2}\right]_{1}^{2}-\left[\frac{x^{-2+1}}{-2+1}\right]_{1}^{2} =\left[\frac{x^{2}}{2}\right]_{1}^{2}+\left[\frac{1}{x}\right]_{1}^{2}\) \(=\left[\frac{4}{2}-\frac{1}{2}\right]+\left[\frac{1}{2}-\frac{1}{1}\right] =\frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1\)
86372
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x=\)
1 \(\frac{\pi}{4}\)
2 \(\frac{3 \pi}{4}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{3}\)
Explanation:
(C) : \(I=\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \sin ^{2}(-x) d x \Rightarrow I=2 \int_{0}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \frac{1-\cos 2 x}{2} d x \Rightarrow I=\int_{0}^{\pi / 2} 1-\cos 2 x d x\) \(I=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\frac{\pi}{2}\)
MHT CET-2020
Integral Calculus
86383
\(\int_{0}^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}\) is
1 \(\mathrm{e}^{1000}-1\)
2 \(\frac{e^{1000}-1}{e-1}\)
3 \(1000(\mathrm{e}-1)\)
4 \(\frac{\mathrm{e}-1}{1000}\)
Explanation:
(C) : Given, \(\int_{0}^{1000} e^{x-[x]} d x=1000 \int_{0}^{1} e^{x-[x]} d x\) \([\) Step function \((\mathrm{x})]=0\) for \((0,1)=1\) for \((1,2)\) and so on \(\}\) \(=1000 \int_{0}^{1} \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}=1000\left[\mathrm{e}^{\mathrm{x}}\right]_{0}^{1}=1000\left[\mathrm{e}^{1}-\mathrm{e}^{0}\right]\) \(=1000(\mathrm{e}-1)\)
UPSEE-2008
Integral Calculus
86401
\(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x=\)
1 \(\frac{5}{3}\)
2 \(\frac{3}{5}\)
3 1
4 -1
Explanation:
(C) : \(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x\) \(=\int_{1}^{2} \frac{x^{3}}{x^{2}} d x-\int_{1}^{2} \frac{1}{x^{2}} d x =\int_{1}^{2} x d x-\int_{1}^{2} x^{-2} d x\) \(=\left[\frac{x^{2}}{2}\right]_{1}^{2}-\left[\frac{x^{-2+1}}{-2+1}\right]_{1}^{2} =\left[\frac{x^{2}}{2}\right]_{1}^{2}+\left[\frac{1}{x}\right]_{1}^{2}\) \(=\left[\frac{4}{2}-\frac{1}{2}\right]+\left[\frac{1}{2}-\frac{1}{1}\right] =\frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1\)
86372
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x=\)
1 \(\frac{\pi}{4}\)
2 \(\frac{3 \pi}{4}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{3}\)
Explanation:
(C) : \(I=\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \sin ^{2}(-x) d x \Rightarrow I=2 \int_{0}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \frac{1-\cos 2 x}{2} d x \Rightarrow I=\int_{0}^{\pi / 2} 1-\cos 2 x d x\) \(I=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\frac{\pi}{2}\)
MHT CET-2020
Integral Calculus
86383
\(\int_{0}^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}\) is
1 \(\mathrm{e}^{1000}-1\)
2 \(\frac{e^{1000}-1}{e-1}\)
3 \(1000(\mathrm{e}-1)\)
4 \(\frac{\mathrm{e}-1}{1000}\)
Explanation:
(C) : Given, \(\int_{0}^{1000} e^{x-[x]} d x=1000 \int_{0}^{1} e^{x-[x]} d x\) \([\) Step function \((\mathrm{x})]=0\) for \((0,1)=1\) for \((1,2)\) and so on \(\}\) \(=1000 \int_{0}^{1} \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}=1000\left[\mathrm{e}^{\mathrm{x}}\right]_{0}^{1}=1000\left[\mathrm{e}^{1}-\mathrm{e}^{0}\right]\) \(=1000(\mathrm{e}-1)\)
UPSEE-2008
Integral Calculus
86401
\(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x=\)
1 \(\frac{5}{3}\)
2 \(\frac{3}{5}\)
3 1
4 -1
Explanation:
(C) : \(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x\) \(=\int_{1}^{2} \frac{x^{3}}{x^{2}} d x-\int_{1}^{2} \frac{1}{x^{2}} d x =\int_{1}^{2} x d x-\int_{1}^{2} x^{-2} d x\) \(=\left[\frac{x^{2}}{2}\right]_{1}^{2}-\left[\frac{x^{-2+1}}{-2+1}\right]_{1}^{2} =\left[\frac{x^{2}}{2}\right]_{1}^{2}+\left[\frac{1}{x}\right]_{1}^{2}\) \(=\left[\frac{4}{2}-\frac{1}{2}\right]+\left[\frac{1}{2}-\frac{1}{1}\right] =\frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1\)
86372
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x=\)
1 \(\frac{\pi}{4}\)
2 \(\frac{3 \pi}{4}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{3}\)
Explanation:
(C) : \(I=\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \sin ^{2}(-x) d x \Rightarrow I=2 \int_{0}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \frac{1-\cos 2 x}{2} d x \Rightarrow I=\int_{0}^{\pi / 2} 1-\cos 2 x d x\) \(I=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\frac{\pi}{2}\)
MHT CET-2020
Integral Calculus
86383
\(\int_{0}^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}\) is
1 \(\mathrm{e}^{1000}-1\)
2 \(\frac{e^{1000}-1}{e-1}\)
3 \(1000(\mathrm{e}-1)\)
4 \(\frac{\mathrm{e}-1}{1000}\)
Explanation:
(C) : Given, \(\int_{0}^{1000} e^{x-[x]} d x=1000 \int_{0}^{1} e^{x-[x]} d x\) \([\) Step function \((\mathrm{x})]=0\) for \((0,1)=1\) for \((1,2)\) and so on \(\}\) \(=1000 \int_{0}^{1} \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}=1000\left[\mathrm{e}^{\mathrm{x}}\right]_{0}^{1}=1000\left[\mathrm{e}^{1}-\mathrm{e}^{0}\right]\) \(=1000(\mathrm{e}-1)\)
UPSEE-2008
Integral Calculus
86401
\(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x=\)
1 \(\frac{5}{3}\)
2 \(\frac{3}{5}\)
3 1
4 -1
Explanation:
(C) : \(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x\) \(=\int_{1}^{2} \frac{x^{3}}{x^{2}} d x-\int_{1}^{2} \frac{1}{x^{2}} d x =\int_{1}^{2} x d x-\int_{1}^{2} x^{-2} d x\) \(=\left[\frac{x^{2}}{2}\right]_{1}^{2}-\left[\frac{x^{-2+1}}{-2+1}\right]_{1}^{2} =\left[\frac{x^{2}}{2}\right]_{1}^{2}+\left[\frac{1}{x}\right]_{1}^{2}\) \(=\left[\frac{4}{2}-\frac{1}{2}\right]+\left[\frac{1}{2}-\frac{1}{1}\right] =\frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1\)
86372
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x=\)
1 \(\frac{\pi}{4}\)
2 \(\frac{3 \pi}{4}\)
3 \(\frac{\pi}{2}\)
4 \(\frac{\pi}{3}\)
Explanation:
(C) : \(I=\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \sin ^{2}(-x) d x \Rightarrow I=2 \int_{0}^{\pi / 2} \sin ^{2} x d x\) \(I=2 \int_{0}^{\pi / 2} \frac{1-\cos 2 x}{2} d x \Rightarrow I=\int_{0}^{\pi / 2} 1-\cos 2 x d x\) \(I=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\pi / 2} \Rightarrow I=\frac{\pi}{2}\)
MHT CET-2020
Integral Calculus
86383
\(\int_{0}^{1000} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}\) is
1 \(\mathrm{e}^{1000}-1\)
2 \(\frac{e^{1000}-1}{e-1}\)
3 \(1000(\mathrm{e}-1)\)
4 \(\frac{\mathrm{e}-1}{1000}\)
Explanation:
(C) : Given, \(\int_{0}^{1000} e^{x-[x]} d x=1000 \int_{0}^{1} e^{x-[x]} d x\) \([\) Step function \((\mathrm{x})]=0\) for \((0,1)=1\) for \((1,2)\) and so on \(\}\) \(=1000 \int_{0}^{1} \mathrm{e}^{\mathrm{x}-0} \mathrm{dx}=1000\left[\mathrm{e}^{\mathrm{x}}\right]_{0}^{1}=1000\left[\mathrm{e}^{1}-\mathrm{e}^{0}\right]\) \(=1000(\mathrm{e}-1)\)
UPSEE-2008
Integral Calculus
86401
\(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x=\)
1 \(\frac{5}{3}\)
2 \(\frac{3}{5}\)
3 1
4 -1
Explanation:
(C) : \(\int_{1}^{2} \frac{x^{3}-1}{x^{2}} d x\) \(=\int_{1}^{2} \frac{x^{3}}{x^{2}} d x-\int_{1}^{2} \frac{1}{x^{2}} d x =\int_{1}^{2} x d x-\int_{1}^{2} x^{-2} d x\) \(=\left[\frac{x^{2}}{2}\right]_{1}^{2}-\left[\frac{x^{-2+1}}{-2+1}\right]_{1}^{2} =\left[\frac{x^{2}}{2}\right]_{1}^{2}+\left[\frac{1}{x}\right]_{1}^{2}\) \(=\left[\frac{4}{2}-\frac{1}{2}\right]+\left[\frac{1}{2}-\frac{1}{1}\right] =\frac{3}{2}-\frac{1}{2}=\frac{2}{2}=1\)
