1) | xe3*dx хе 2) | x2 sin x dx 3) x sin 2x dx

Q: 1. Integrate: [x³ sin(2x) dx.

A: As per Bartleby guideline, only the first question has been answered. If you want to get the answer…

Q: sin 3x cos 3x (cos 3x + 1)² dx 19) f 22) fex√3-4e2x dx 20) f ITV In x dx 21) f x√4 lnx-1 23)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 2. 1= j tan(4x³) ,sec²(4x²)xdx dx 1- I=31-2x) (In( 5. I= 3. 1= ftan? (3x) sec? (3x)dx

A: Here we have to find the value of the following integrations. Note :   As per our company guidelines…

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

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Q: 7/4 2 sec(6x)dx sin(r In(x)) 13x 1

A: (b)

Q: 2. Show that P(1-x)° dx= x'(1-x) dx. %3D 0/

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Q: - sinx) dx 2. 1. (COSX (ax' +b)*x dx 2. 2/3

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Q: 1 Evaluate [ (x³ +cos(3x))dx 2

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

Q: 7 show that: sech x dx = Sin (tankx). √1-u² multiply note: * & Sim'u = dx * you Cam سال dx sechx by…

A: Differentiate rhs

Q: 10. Find , if y = dy 1-cosx dx 1+cosx

A: Given  y = (1- cosx) /(1 +cosx) To find dy/dx

Q: (a) If y = xV², then dy dx (b) If y = (2 + sin 3x)", then d (2 + sin 3x)" = T(2 + sin 3x)=-1(cos 3x)…

A: We have to differentiate the given functions: a If y=x2, then dydx=2x2-1 b If y=2+sin 3xπ,then…

Q: 9. Integrate: Ja (x- 1) cos(x? - 2x) dx

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

Q: 1- d/dz [cosz] = -sin z

A:

Q: 2. [2x*tan y – ysin (2) – 1] dx + [xsin (2) + x*sec*y] dy =

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Q: ¥2/3 6. cos'x sin x dx 5.

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Q: 9.) JE sin°(x)cos*(x)dx 11.) F sin°(x)cos²(x)dx 2

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: d Verify tanh - = X 1- x2 dx

A: Inverse hyperbolic tangent function can be expressed in logarithm funtion. Write tanh-1x=12ln1+x1-x…

Q: 4- Find dy/dx for: d- y = /1+ sin³ (x5) e- y = 2(Va – 1)( f- y = cos³{sin(2x*)} x-2 x²+3x' %3D

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Q: 2 sin 2x dx Tcost x + 4 cos² x – 2

A: We will answer the first question since exact one was not specified. Please resubmit the question…

Q: 24. Evaluate: S(cos 2x) esin 2x dx A. -e sinzx + C esinzx +C C. esinzx + C esinzx D. + C В.…

A: using substitution method to solve integration

Q: 21. Integrate x cos(2x² + 7)dx. sin (2x²+7) 4(x²+7) B. sin(2x²+7)+c A. +C C. D. 4 4 sin(2x² +7)+c…

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Q: x* cos 3.xdx 2 fr' cos 3.xdx sin 3xsin 2.xdxr

A: Solving

Q: 2. The suitable Trignometric substitution that solves fx(x-2) dx is (x-1)=sin 0 x-1)3sec0 (x-1)=tan…

A: We can find the answer as below.

Q: Step by step

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Q: 2. +4y= sin(3t), y(0)=2, y'(0) = 0

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Q: 19. -π/2 21. -π/4 23. [³5e* dx sin 0 de cos x dx sec² 0 do (2x secx tan x) dx 20. rπ/3 22. 24. [¹-dx

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Q: 3. J sin53x dx. 4. J tan 2x sec22x dx.

A: ∫ax+bn dx=1aax+bn+1n+1+C

Q: [2 2 sinx cosx + 3cosx7 2 cosh 3 X 1 D²+ 4 D3 D³ -27

A:

Q: 1. Evaluate 1 (a) [ In(2x + 2)dx (b) [ sec'3x tan 3x dx

A: (1)(a) Use the method of substitution to solve the definite integral.…

Q: roo sin(2) Compute (4) dx. fo x(x²+4)

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Q: Show that sin(x^2) ≤ sin x for 0 ≤ x ≤ 1. Deduce that sin(x^2) dx ≤√2/2 on the invterval [0, π/4]

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Q: 211 3- sin nx dx [1-cos2nx]dx 1t, where n %3D

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Q: d 1+ tanhx - Show that dx V - tanh x

A: To show: ddx1+tanhx1-tanhx4=12ex2 Simplfiying 1+tanhx1-tanhx,…

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: Exercises to solve: 1. fx² In(x + 1) dx 2. (x2 + x – 2).cos4x. dx 3. Se-*sinxdx 4. S sinh2x.cosh5x…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 2.1 Given that 1 sinh x show that dy cosh a. dx

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

A: a) Given integral is ∫11+sinxdx

Q: ,1 x² - 4x⁹+ A) True B False sin(x) (1+x²)² |dx=0

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Q: х2 9 dy + sinx - у %3D0 dx

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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: +15 - If y = cos-1(x) , Prove that dy -1 %3D dx /1-x²

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Q: dr 21. (1+x²)/16 – (tan-l r)² dx e2x+2+2 22. 8. dx x2 - 6x + 25 23. 6. 24. dr J (2 - r)V - 4x +3 dr…

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Q: 211 3- sin nx dx [1-cos2nx]dx 1t, where n integers.

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Q: 1 Integrate: S tan*÷x dx 3

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Q: 6. x sec x tan x dx

A: Let's find.

Q: (7) (4+sin-15x)° dx V1-25x2 3. dx (8) J Ja +x Vx

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Q: 16. √(16-9x²) dx 17. dx/ (9x² + 4) 18. sin³ x cos x dx 19. f (secs x) dx/csc x 20. Stan² x dx

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Q: 9. [sin Ssin' cos–dx 3 3

A: (9) To evaluate ∫sin5x3cosx3dx Substitute sinx3=u and cosx3dx=3du. Therefore,…

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.