Integrate the following function (a) `int_(o)^(2) 2t dt` (b) `int _(pi//6)^(pi//3) sin x dx` (c) `int _(4)^(10)(dx)/(x)` (d) `int _(o)^(pi) cos x dx`

Question

Integrate the following function (a) `int_(o)^(2) 2t dt` (b) `int _(pi//6)^(pi//3) sin x dx` (c) `int _(4)^(10)(dx)/(x)` (d) `int _(o)^(pi) cos x dx`

Integrate the following function
(a) `int_(o)^(2) 2t dt`
(b) `int _(pi//6)^(pi//3) sin x dx`

(c) `int _(4)^(10)(dx)/(x)`
(d) `int _(o)^(pi) cos x dx`
(e) `int _(1) ^(2)(2t -4) dt`

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

(a) `vec0` (b) 1 (c) 0 (d) `(1)/(x)` (e) `-(1)/(2)`
(f) `int_(0)^(pi//4)sin 2x dx=[-(cos 2x)/(2)]_(0)^(pi//4)=-(1)/(2)(cos(2pi)/(4)-cos 0) =-(1)/(2)(0-1)=(1)/(2)`
(g) `(d^(2))/(dx^(2))(e^(3x))=(d)/(dx)(e^(3x)(d)/(dx)(3x))=(d)/(dx)(3e^(3x))=3e^(3x).3=9e^(3x)`
(h) `|2hati-hatj-5hatk|=sqrt(2^(2)+(-1)^(2)+(-5)^(2))=sqrt(30)`
(i) `(d^(2))/(dx^(2))(4x^(3)-3x^(2)+2x+1)=(d)/(dx)(12x^(2)-6x+2)=24x-6`
(k) `int_(-1)(+1)t^(-3dt)=[(t^(-2))/(-2)]_(-1)^(+1)=[-(1)/(2t^(2))]_(-1)^(+1)=(-(1)/(2(1)^(2)))-(-(1)/(2(-1)^(2)))=-(1)/(2)+(1)/(2)=0`
(l) `hati.(hatjxxhatk)+hatj.(hatkxxhati)+hatk.(hatixxhatj).=hati.hati+hatj.hatj+hatk.hatk=1+1+1=3`