Part 1: 1 1. S *(In x)3 dx 2. Se3x cos x dx 3. S(1 + x²)e-*dx 4. So x²e*dx

Q: 1 2 de 4-cose %3D II

A: Given,          ∫0π214-cosθdθ

Q: 3 S² x³ cos x ²dx 6.

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Q: 11, 0-20² 2. 3. चं 5. T S cos 2x dx sin(²³) cos²(:-) / dx 2x-1 4x dx dx 6. 7. 8. S [² (x² + 0²³) dx…

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Q: 2 tan(4x-1)-3 cos² (4x-1) If we let u = 4x – 1 for the dx, what will be the resulting integrand?…

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Q: 18) e2x = sin(x + 3y)

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Q: 1. [ (6x² – 2x+3)dx sin(2x)dx 3 2. 3. S2 8 dx X+2

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Q: B. By miscellaneous substitution (VE+1)dx dx 1. S 2. S: 3 cos x+2 sin x-5 XA +

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Q: 3 tan-(x)dx O V3 - In (8) V3 TL In (4) 2/3 2.

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Q: f(x)=X +Sin -1 く(-2)- 4-て+Sini-て)-\no9651 3(x)=てメ+ cosx ずし-て) ニ -4ACOS(-て)-3 (x)-2-らinx f(-2):…

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Q: of (e² 2-X Sin (2y)

A: Given, f= e2-xsin-1(2y) To find, df/dy.

Q: Evaluate D(sinx – 4e2) D³(sinx cos x-32 e В -cos x-32e2 C) -sin x-32 e2x (D sin x-32 ex A

A: The objective is to evaluate, D3sinx-4e2x

Q: 1. sin x sin ydx + cos x cos ydy = 0 2. 3 ydx = (3x + 1)dy 3. (х — 2 у)dx + (2х+ у)dy %3D 0

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Q: 1. Integrate: e3x cos(x) dx

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Q: B: 3:Use Gamma & Beta properties to evaluate : f (In x) 3 dx

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Q: 3+9sin' x Evaluate dx using suitable technique. cos x

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Q: sin(4x) cos(2.x) dx n (2x) sin (6x) cos (2x) cos 2. ()= 4 'cos (2x) cos (10x sin (2x) trts 10 (e)…

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Q: 1 Evaluate [ (x³ +cos(3x))dx 2

A: The given integral is ∫x3+12cos3xdx.

Q: - / sin 8x cos 5x dx = 1/6 cos3x + 1/26 cos13x + C ) -1/6 cos3x + 1/26 cos13x + C ) -1/6 cos3x -…

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Q: d sin x 1 dt| = dx et + 5 cO X 1 1 1 (A) esinx + 5 (B) e* + 5 (C) (esinx + 5) cos x (D) e sin x

A: We know that integration power rule is given as - Integration of xn is nxn-1.   Here i attached the…

Q: 1. f/² sin () cos (+) sin (²) d dx (Consider initially the first 2 terms)

A: We have to use the formulas for 2sin(A)cos(B)-sin(A+B)+sin(A-B) and 2sin(A)sin(B)=cos(A-B)-cos(A+B).…

Q: To evaluate 3cosx -1 1- sin?x dx, the integrand can be transformed to A J(3 cos x - sec²x) dx ® S(3…

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Q: 3e"dx 3x 1. 3x – 16 2. x 10 – 3x°dx 3. / In (cos®) an@ de 3.

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Q: 2. 1- x2dx cos(x + 1) dx 3.

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Q: (a) Show that cos(x²) > cos x for 0 ;. 1

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Q: 4) f(x)=- - T <x< T Ans. 4 cos(x) +cos(2.x) - 4 cos(3x) +... 12

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Q: S cos ( x^3 ) dx = * sin ( x^3) + c 3x^2 sin (x^3 ) + c - sin (x^3 ) + c - 3x^2 sin ( x^3) + c none…

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Q: sin(6x) cos(4x) dx = sin (2x) sin (6x) (b) 글(s сов (2х) cos (6x) 2 o+( cos (2x) cos (10x) sin (2x)…

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Q: -2/√√/3 dx x√√x²-1 • [²/²2 (x + 12/²3). dx sin² π/6 28. • √2 30.

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Q: 4x /² sin () cos (+) sin (²) dx (Consider initially the first 2 terms) 3

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Q: -2.1 1- e* Use Simpson's rule to approximate -dx. x² + 2x – 1 1.2

A: This is a problem of numerical Integration.

Q: cos x 3. f sin³x dx (2 + sin?x) + C 3 4. S* x sin 2x dx 1-9x2 5. S sin-1 3x dx + x sin-1 3x + C 3

A: Q - 3

Q: 9-x 4. Evaluate ſ, S sin(x² + y²) dy dx

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Q: Question 8 Integrate the function. dx V9 - x2 COs:1 X+C 2 cos x+ sin x+C D sin *-C Question 9 413

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Q: 1. Evaluate f sin*(2x + 3)cos (2x + 3) dx sin(2x + 3) put = t 2. Evaluate f dx Put x2 + 5 = t x2+5…

A: Explanation of the solution is given below.....

Q: 4 sin' 2x cos 4x dx 4.

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Q: sin(4x) cos (2x) dx = +(st)(9) (e) None sin sin (con (2x) Cos сов (10х) sin (2x) sin (10x) (e)( (Cо…

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Q: Vsin x.cos x 1+ 2.ln x 3y (аns.: 4

A: Given                   y43= sinx cosx1+2.lnx

Q: 1 a) 1+ sin(x) xp- Vtan (x)dx

A: a) Given integral is ∫11+sinxdx

Q: 9. Find the value of k so that (2y sin x cos x - y+ 2y²e")dx - (x- sin? x+ kxye*)dy = becomes exact.…

A: To find the  value such that given equation is exact  differential equation.

Q: =1+ cos (x) x2 + 12x + 20 -dx 1 + cos² (x) 0. At what value of t does the local max of f(t) occur?

A: Given that f(t)=∫0tx2+12x+201+cos2 xdx To find the local maximum of f(t) at t:

Q: 1. f/² sin (1) cos (4) Sir sin (²) dx (Consider initially the first 2 terms)

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Q: Example 1 Find dy/dx, if a. y=sin(x² + 3x + 1) b. y = cos² (2x-1)

A: I have stated the used formulas in the solution

Q: (1) ſ [ (x³ + 2 + 7 ) dx (3) cos³x dx (2) √ x³ (2) x3 e3x+1 dx (4) sin x sin 2x dx

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Q: 7/2 in° x cos' xr dx 4

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Q: (4) find dy if 3って - ソ= e* (2Co83x +3Sin3x) e = Sin (x+3y).

A: Given 1. y=e2x2cos3x+3sin3x 2. e2x=sinx+3y

Q: 3) e4x = sin(3x + 4y)

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Q: (x2+ 1)(x tan 8 dx 81. dx 3) (x+2)3 dx (sin (sin (e*))) 83. (In (sec x tan x)) dr dx xex P (sin…

A: We’ll answer the first question since the exact one wasn’t specified.Please submit a new question…

Q: 5.S Vsin 2x . cos2 x dx

A: Integration is a process of adding up infinitesimally small pieces of an area. There are mainly 2…