Consider I1-0-4 xe-xdx- I2-0-42 xe-xdx- I3-0-42 xe-x2dx- and I4-0-43 xe-x2dx-Then correct sequences is-are

When nbsp; x∈ ( 0,π4 nbsp; ), nbsp; x ∈ (1, ∞ ) ⇒ x lt; ^2 x lt; ^3 x and x^2 lt; x , x^2 gt; x ⇒ I_4 gt; I_3 gt; I_2 gt; I_1 ...

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Integration of Piecewise Continuous Functions

Consider I1=∫...

Question

Consider $I_{1} = π / 4 \int 0 (cot x) e^{- x} d x,$ $I_{2} = π / 4 \int 0 ({cot}^{2} x) e^{- x} d x,$ $I_{3} = π / 4 \int 0 ({cot}^{2} x) e^{- x^{2}} d x,$ and $I_{4} = π / 4 \int 0 ({cot}^{3} x) e^{- x^{2}} d x .$
Then correct sequence(s) is/are

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I_{1} = \int_{0}^{\frac{π}{4}} e^{x^{2}} d x, I_{2} = \int_{0}^{\frac{π}{4}} e^{x} d x, I_{3} = \int_{0}^{\frac{π}{4}} e^{x^{2}} cos x d x, I_{4} = \int_{0}^{\frac{π}{4}} e^{x^{2}} sin x d x

I_{2} > I_{1} > I_{3} > I_{4} .

x (0, 1), x > x^{2}

sin x > cos x .

I_{1} = \int_{0}^{π / 2} {sin}^{4} x d x

I_{2} = \int_{0}^{π / 2} {cos}^{6} x d x

I_{3} = \int_{0}^{π / 2} {sin}^{8} x d x

I_{4} = \int_{0}^{π / 2} {cos}^{2} x d x

I_{1}, I_{2}, I_{3}, I_{4}

I_{n} = \int_{0}^{π / 4} {tan}^{n} (x) d x

I_{2} + I_{4}, + I_{3} + I_{5}, + I_{4} + I_{6}, . . .

I_{1} = \int_{0}^{π / 2} \frac{x}{sin x} d x

I_{2} = \int_{0}^{π / 2} \frac{{tan}^{- 1} x}{x} d x,

\frac{I_{1}}{I_{2}} =

I_{1} = \int_{0}^{\frac{π}{4}} x^{2008} (t a n x)^{2008} d x, I_{2} = \int_{0}^{\frac{π}{4}} x^{2009} (t a n x)^{2009} d x a n d I_{3} = \int_{0}^{\frac{π}{4}} x^{2010} (t a n x)^{2010} d x

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