Calculate F (6) given that F (3) = 7 and F' (x) = x². Hint: Express F (6) – F (3) as a definite integral. (Use symbolic notation and fractions where needed.)

Q: Use the substitution x = 4 tan(0) to evaluate the indefinite integral 17dx x2 Vx2 + 16 + C

A:

Q: Evaluate the indefinite integral. x + 1 da x² + 2x + 2 + C

A:

Q: What is the definite integral of a function ƒ over a closed interval3a, b4 ? When can you be sure it…

A: The definite integral of a function f(x) over a closed interval [a,b] is a real number  denoted by…

Q: Find the indefinite integral. Solution in handwritten. 1. f =dx /1+x4

A:

Q: Evaluate the integral. (Use C for the constant of integration.) x2 - 81 xp

A: Let,I=∫x2-81x3dxI=∫x2-81.1x3dxI=x2-81∫1x3dx-∫∫1x3dxddxx2-81dxI=x2-81∫x-3dx-∫∫x-3dxddxx2-81dxI=x2-81x…

Q: Find the Absolute Extremum of F(x)=2sinx-cos2x over [0,2pi]

A: Consider the following:  Differentiate the function and find critical point:…

Q: Calculate the definite integral of x^2 + 4x approaches from 1 to 3.

A:

Q: Evaluate the definite integral.

A:

Q: Express the limit as a definite integral on the given interval. im. (4(x;") - 3x;JAx, [2, 7] i = 1 7…

A: By limit definition of a definite integral, limn→∞∑i=1nfxi*∆x=∫ab f(x)dx  where ∆x=b-an and…

Q: Determine the following integral: 3x + 3 Jx2 + 7x + 12 as a function of x.

A: First we will factor the denominator of the integrant and then by doing the partial fraction we will…

Q: x2 – x dx -1

A: To evaluate the definite integral -1 to 1.

Q: Evaluate the indefinite integral f/2+ /2 + x*dx =

A: Given :-∫x3 2+x4 dxsolution:-using integration by u-substitutionlet,2+x4=udifferentiate both…

Q: Find the indefinite integral

A: To solve the integral, use the substitution method and consider x4-4 as u. Differentiate both sides…

Q: Evaluate the definite integral. sech2(8x – 1) dx 1/8

A:

Q: Evaluate the indefinite integral: (x² – 3)*(2x) dx -

A:

Q: Evaluate the integral from -n/2 to +t/2 of x² sin x.

A: Given integral

Q: Find the indefinite integral by making the substitution x = 9 sin e. 6. V81-x2 1 3/2 +C 81 V81-…

A: Option b is correct

Q: Find the indefinite integral of f(x) = x^1/3

A:

Q: Find the indefinite integral of the function: f(x) = 1/(x+2)

A: The given function is  f(x)=1x+2

Q: Express the limit as a definite integral on the given interval. I [2(x;)3 - 7x;1Ax, [2,9] lim i = 1…

A: Answer is mentioned below

Q: Express the integrand as a rational function and then evaluate the integral. x - 4 dx Jo x² – 5x + 6

A:

Q: Use the substitution x 3 tan(0) to evaluate the indefinite integral 91dx x2 Vx2 + 9 + C

A:

Q: Evaluate the integral. (Use C for the constant of integration.) x² - 64 dx - +8

A:

Q: Evaluate the integral. (Use C for the constant of integration.) x² - 16 dx +8

A: We have to find the integral

Q: 4. Evaluate the definite integral by using substitution. x-1 dx (x+ 2)2/3 -1

A: given the definite integral is∫-16 x-1(x+2)23dx

Q: L, rVx* + 1 dx -3

A: we have to find out the integration

Q: Find the indefinite integral. xx-6x+9 f ***) dx 3.

A:

Q: 15 Ső 0 X √x+1 ах

A: We have to solve the given definite integral

Q: 3x²-2 do 3 2x - 4x

A: Here we will solve the given integral,

Q: Find the indefinite integral by making the substitution x- 7tan e. dx 49+x²

A: See attached file for a step by step explanation.

Q: Зx2 + 4х + 1 dx ). *3

A:

Q: Find the indefinite integral f x5e 2x°dx .

A:

Q: – dx x Inx x²+1

A:

Q: 3x2 + 2x +1 dx x2

A: See the attachment

Q: sech(In x) 9) dx

A: ∫15 sechln xxdx

Q: Find the indefinate integral for f(x) = (x^1/2)+(1/x)-9e^x

A: Given,

Q: Evaluate the definite integral of f(x)=3x^2+2x+ x^(1/2) over the interval [0,1]

A:

Q: dx 1. -1 V1 – 2x – x²

A: Explained below

Q: Evaluate the indefinite integral:  x3ln(x2-1)dx

A: Given integral By using the following substitution, we get

Q: Find the indefinite integral by using the substitution x = 2 tan(0). (Use C for the constant of…

A: Given ∫2x24+x22dx and x=2tanθ

Q: Evaluate the definite integral. dx x(1+ In æ)

A: Explained below

Q: Evaluate the indefinite integral.

A: It is required to evaluate the indefinite integral:

Q: dr (x² + 1)*

A:

Q: Find the indefinite integral. (Use C for the constant of integr √(x-8 - 8)² dx

A:

Q: Evaluate the definite integral. 2x+ dx x² +x* +1

A: The given integral is ∫012x3+xx2+x4+1dx. Solve the given integration by integration by substitution…

Q: Evaluate the integral. (Use C for the constant of integration.) x2 xp V 81 - x2

A:

Q: Show, using the formula for the derivative of the inverse of a function,that the derivative of…

A: The solution of above question is written below.

Q: Vx2 – 4 dx x2

A:

Q: Evaluate the definite integral. 5x2 + 1 dx = x2

A:

Q: Use the Change of Variables Formula to evaluate the definite integral. 6 In x dp- 1

A:

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.