Integrals

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New answer posted•a year ago

Integrals Class 12th

$\underset{x \to \frac{π^{-}}{2}}{l i m} \frac{\int_{x^{3}}^{{(\frac{π}{2})}^{2}} c o s t^{1 / 3} d t}{{(x - \frac{π}{2})}^{2}}$

8 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

$\underset{h \to 0}{l i m} \frac{\int_{(\frac{π}{2} - h)}^{{(\frac{π}{2})}^{3}} c o s (t^{1 / 3}) d t}{h^{2}}$

$= \underset{h \to 0}{l i m} \frac{0 + 3 {(\frac{π}{2} - h)}^{2} c o s (\frac{π}{2} - h)}{2 h}$

$= \underset{h \to 0}{l i m} \frac{3 {(\frac{π}{2} - h)}^{2} s i n h}{2 h}$

$= \frac{3 π^{2}}{8}$

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New answer posted•a year ago

Integrals Class 12th

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{8 \sqrt[]{2} c o s x}{(1 + e^{s i n x}) (1 + s i n^{4} x)}$ dx = ap + b log $(3 + 2 \sqrt[]{2})$

11 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{8 \sqrt[]{2} c o s x}{(1 + e^{s i n x}) (1 + s i n^{4} x)}$ dx

$= \int_{0}^{\frac{π}{2}} {(\frac{8 \sqrt[]{2} c o s x}{(1 + e^{s i n x}) (1 + s i n^{4} x)} + \frac{8 \sqrt[]{2} c o s x}{(1 + e^{- s i n x}) (1 + s i n^{4} x)})} d x$

$= 8 \sqrt[]{2} \int_{0}^{\frac{π}{2}} \frac{c o s x}{1 + s i n^{4} x} d x$

Let sin x = t

$I = 8 \sqrt[]{2} \int_{0}^{1} \frac{d t}{1 + t^{4}}$

$= 4 \sqrt[]{2} \int_{0}^{1} \frac{(1 + \frac{1}{t^{2}}) - (1 - \frac{1}{t^{2}})}{t^{2} + \frac{1}{t^{2}}} d t$

$= 4 \sqrt[]{2} \int_{0}^{1} \frac{(1 + \frac{1}{t^{2}}) d t}{{(t - \frac{1}{t})}^{2} + 2} - 4 \sqrt[]{2} \int_{0}^{1} \frac{(1 - \frac{1}{t^{2}}) d t}{{(t + \frac{1}{t})}^{2} - 2}$

$= 4 \sqrt[]{2} \cdot \frac{1}{\sqrt[]{2}} {(t a n^{- 1} \frac{t - \frac{1}{t}}{\sqrt[]{2}})}_{0}^{1} - 4 \sqrt[]{2} \cdot \frac{1}{2 \sqrt[]{2}} {[l o g | \frac{t + \frac{1}{t} - \sqrt[]{2}}{t + \frac{1}{t} + \sqrt[]{2}} |]}_{0}^{1}$

$= 2 π - 2 l o g | \frac{2 - \sqrt[]{2}}{2 + \sqrt[]{2}} |$ &nbs

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New answer posted•a year ago

Integrals Class 12th

If $\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, (a > 2) a n d [x]$ denotes the greatest integer $\leq x,$ then $\int_{a}^{- a} (x + [x]) d x$ is equal to__________.

7 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

$\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, a > 2$

$\int_{- a}^{0} (- 2 x + 2) d x + \int_{0}^{2} (x - x + 2) d x + \int_{2}^{a} (2 x - 2) d x = 22$

$\Rightarrow 2 a^{2} + 2 = 20 \Rightarrow a^{2} = 9 \Rightarrow a = 3$

$∴ \int_{3}^{- 3} (x + [x]) d x = - \int_{- 3}^{3} (2 x - {x}) d x = - \int_{- 3}^{3} 2 x d x + 6 {\int_{0}^{1} x d x = 6 . \frac{x^{2}}{2} |}_{0}^{1} = 3$

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New answer posted•a year ago

Integrals Class 12th

The value of the integral

$\int \frac{s i n θ s i n θ (s i n^{6} θ + s i n^{4} θ + s i n^{2} θ) \sqrt[]{2 s i n^{4} θ + 3 s i n^{2} θ + 6}}{1 - c o s 2 θ} d θ$ (where c is a constant of integration)

6 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let sin = t

$\int \frac{s i n θ (2 s i n θ . c o s θ) (s i n^{6} θ + s i n^{4} θ + s i n^{2} θ) \sqrt[]{2 s i n^{4} θ + 3 s i n^{2} θ + 6}}{2 s i n^{2} θ}$ d

sin = t

cos . d = dt

$\int \frac{u^{1 / 2}}{12} d u = \frac{u^{3 / 2}}{18} + C = \frac{{(2 t^{6} + 3 t^{4} + 6 t^{2})}^{3 / 2}}{18} + C = \frac{{(2 s i n^{6} θ + 3 s i n^{4} θ + 6 s i n^{2} θ)}^{3 / 2}}{18} + C$

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New answer posted•a year ago

Integrals Class 12th

The value of $\underset{x \to 1}{l i m} \frac{(x^{2} - 1) s i n^{2} (π x)}{x^{4} - 2 x^{3} + 2 x - 1}$ is equal to:

9 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

$x^{4} - 2 x^{3} + 2 x - 1 = {(x - 1)}^{2} (x^{2} - 1)$

$s i n π x = s i n (π (1 - x)$

$= - s i n (s i n π (x - 1))$

$\underset{x \to 1}{l i m} \frac{(x^{2} - 1) \cdot s i n^{2} π x}{(x^{2} - 1) {(x - 1)}^{2}} = \underset{x \to 1}{l i m} \frac{s i n^{2} (π (x - 1))}{{(x - 1)}^{2}}$

$= \underset{x \to 1}{l i m} \frac{s i n^{2} (π (x - 1))}{{(π (x - 1))}^{2}} \cdot π^{2}$

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New answer posted•a year ago

Integrals Class 12th

Value of $\int_{- π / 2}^{π / 2} \frac{d x}{(1 + t a n^{2} x) (1 + 2^{x})}$ equals

13 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

$\int_{0}^{π / 2} (\frac{1}{(1 + t a n^{2} x) (1 + 2^{x})} + \frac{1}{(1 + t a n^{2} x) (1 + 2^{- x})}) d x$

$\int_{0}^{π / 2} \frac{d x}{(1 + t a n^{2} x)} = \int_{0}^{π / 2} c o s^{2} x d x = \frac{π}{4}$

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New answer posted•a year ago

Integrals Class 12th

∫ cot?¹(1 - x + x² - x³ + x? - x? + .) dx (|x|<1)

7 Views|Posted a year ago

Asked by Raj Pandey

Answered by

Raj Pandey | Contributor-Level 9

a year ago

$\int {c o t}^{- 1} ? (\frac{1}{1 + x}) d x$

$= \int {t a n}^{- 1} ? (1 + x) d x$

$= x {t a n}^{- 1} ? (1 + x) - \int \frac{x}{1 + (1 + x)^{2}} d x; = x {t a n}^{- 1} ? (1 + x) - \int \frac{x}{x^{2} + 2 x + 2} d x$

$= x {t a n}^{- 1} ? (1 + x) - \int \frac{\frac{1}{2} (2 x + 2) - 1}{x^{2} + 2 x + 2} d x = x {t a n}^{- 1} ? (1 + x) - \frac{1}{2} l o g ? (x^{2} + 2 x + 2) + \int \frac{d x}{(x + 1)^{2} + 1}$