Using a suitable substitution, (sin x cos“ x) dx can be 4 (A) - f u*du (B)

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A: The integration can be written as;

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Q: Consider ∫ x sinx cosx dx = f(x) sin2x - g(x) cos2x + C. What is the value of f(1) + g(1)?

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Q: 3G)Ú) Show that the reduction formula for atanx で-u sec"x dx = Sec" 「キu ) Hence use tthe formula to…

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Q: d (sec x) dx Prove that sec x tan x.

A: We have to prove that: ddxsec x=sec x tan x We know the formula of derivatives, dxndx=nxn-1d cos…

Q: Evaluate the integrals

A: We have to evaluate the integral;  13) ∫ln4ln9ex/2 dx14) ∫tanxln(cosx) dx

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Q: What is special about the functions f(x) = sin x - 1 x + 1' x 2 0, and g(x) = 2 tan Vĩ? -1 Explain.

A: Given that functions are f(x)=sin-1x-1x+1, x≥0 and g(x)=2tan-1(x). The graph of the function…

Q: Find f'a) for f(x)= =Sinx 1- Cos X %3D

A: We have to find the derivative of the given function.

Q: cos (In z) sin (In x) dx

A: To Evaluate the integral

Q: d (cos x) H.W.: show that -sin x %3D xp

A: Given: To show ddx(cos x)= -sin x Definition : Derivative of f(x) w.r.t. x is defined as:…

Q: Using appropriate substitution, f sin°(x)cos³ (x)dx is equal to:

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Q: = (a) Evaluate S cosx sinx V1 – sin4x dx. -

A: Given∫cosxsinx1-sin4xdx

Q: Solve f ) = (tanX + Cos x and t Cos X

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Q: 2) Proof that If y = f(x) = = cos x then dx sin x

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Q: sin x dx V2-cos² x

A: Given : ∫(sin x/√2 - cos2 x) dx     ....(1)   Let cos x = t   Differentiating both sides w.r.t. x -…

Q: Consider ! x sinx cosx dx = f(x) sin2x - g(x) cos2x + C. What is the value of f(1) + g(1)?

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Q: ( sin (In 2x) - cos (in 2x))? +x dx 2x

A: Let's find.

Q: Using suitable method, show that 2 tan" (x+2) dx =- – In 2 .

A: We use integration by parts to find integral tan inverse x.

Q: f tanx = 5 and cosx > 0. Sh

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Q: 1. f(x) = cos (-:-)

A: From graph, it is clearlimx→∞cos1x =limx→-∞cos1x = 1

Q: If f(x) = 2x(sina + cos a), find f'(x) = %3D

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Q: Differentiate f(x) = -1 = COS (4x). Use exact values. f'(x) = %3D

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Q: (B) Evaluate tan (3.x +1) sec³(3x +1) dx

A: In this, We find the value of the given integral using tan^x+1=(sec^2)x formula.

Q: f cos² x dx

A:

Q: Use Trigonometric Substitution to evaluate |x'V1-x° dx

A: put x= sin t for trigonometric substitution.

Q: (b) 는(cosx)=-sinx (d) VI+ tan x 1+* Q1) Show that: (a) (1+x) =1+6x (e) (V): *- () %3D

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Q: 57. Extension of Example 8 Show that cos (x) is dx sin (x). 180

A: The objective is to show that ddxcosx° is -π180sinx°. The function cosx° is given in degrees bu the…

Q: Sih X pronf:, sin1x) cosinx)dx Sin (1-h)X

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Q: xy2-cos x sin x y' = ху?-со -cOs x sin У(1-x?)

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Q: [sin(In x) + cos(In x)]* dx 3. xp: 3x/10 In(3x) – (In(3x))²

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Q: Prove the identity. cosh(x) + sinh(x) - e

A: To prove the given identity we uses basic expression for sinh(x) and cosh(x).

Q: du Compute f CSC X u = f(x), then dx sin x – cos x 2 -

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Q: 19. Evaluate using u-substitution: S tan6x sec² x dx

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Q: do sin 0 – cos 0 cos 0

A: The given integral is ∫dθsinθ-cosθ

Q: Suppose f(x, y) = / tan(x) + y and u = (-3,0). Then, Du f(-0.8, 8) =

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Q: (cos(2x) – sin(2x) cos(2x)) cos(x) – sin(x) For -dx%3D +C. What is F (x)?

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Q: Find this second derivative: dz O 2 cos a? – 4x² - sin x² O 4x² cos x² – 2 sin æ² O 82² (cos z²)…

A: Given : Second derivate of sinx2

Q: Evaluate sin (x) e-x dx by use Simpson's rule. h=0.5

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Q: f(x) = /tan-1 (cos x) + In(sec x)

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Q: Evaluate (sin² (sin² x cos x)(1+ sin z cos x) dx.

A: Use substitution method

Q: Calculate tan x ( sec“ x ) dx using u-substitution with: (i) u = tan x (ii) u = sec x

A: I have stated the used formulas in the solution.

Q: cos2x – sin2x cos2x F(x) +C. What is F(M)? For dx%3D cosx – sinx 3

A: A detailed solution is given below

Q: 30. sin x cos(cos x) dx

A: The given integral is ∫sin x coscos x dx. Let u=cos x. Then, du=-sin x dx.

Q: a(cos2x) sin(x-) TC 1. f(x) = tanx 4. What values should a and b take for the function fx) to be…

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