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Simora [160]
2 years ago
8

The indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Th

en find it without using substitution. Check that your answers are equivalent. 6x^5(x^6-2)dx
Mathematics
1 answer:
Fantom [35]2 years ago
7 0

Answer:

∫6x^5(x^6-2)\,dx = \frac{1}{2}(x^6-2)^2+C

Step-by-step explanation:

To find:

∫6x^5(x^6-2)\,dx

Solution:

Method of substitution:

Let x^6-2=t

Differentiate both sides with respect to t

6x^5\,dx=dt

[use (x^n)'=nx^{n-1}]

So,

∫6x^5(x^6-2)\,dx = ∫ t\,dt = \frac{t^2}{2}+C_1 where C_1 is a variable.

(Use ∫t^n\,dt=\frac{t^{n+1} }{n+1} )

Put t=x^6-2

∫6x^5(x^6-2)\,dx = \frac{1}{2}(x^6-2)^2+C_1

Use (a-b)^2=a^2+b^2-2ab

So,

∫6x^5(x^6-2)\,dx = \frac{1}{2}(x^6-2)^2+C_1=\frac{1}{2}(x^{12}+4-4x^6)+C_1=\frac{x^{12} }{2}-2x^6+2+C_1=\frac{x^{12} }{2}-2x^6+C

where C=2+C_1

Without using substitution:

∫6x^5(x^6-2)\,dx = ∫6x^{11}-12x^5\,dx = \frac{6x^{12} }{12}-\frac{12x^6}{6}+C=\frac{x^{12} }{2}-2x^6+C

So, same answer is obtained in both the cases.

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Math workshops and final exams: The college tutoring center staff are considering whether the center should increase the number
oksian1 [2.3K]

Answer:

Option D

Step-by-step explanation:

The response variable is also known as the dependent variable. It depends on another factor which is the independent variable to cause change/ response to it.

In this study, they want to test if an increase in the number of Math workshop which is the independent variable will cause a change in the final exam scores which is the dependent variable. The response to this inquiry is thus the final exam score which is also known as the response variable.

7 0
3 years ago
The Math Counts Club had a party at their school. There were 20 cookies and 40 slices of pizza to be shared equally. Each studen
pychu [463]

Answer:

  • 2, 4, 10 or 20 students

Step-by-step explanation:

<u>Prime factors:</u>

  • 20 = 2*2*5
  • 40 = 2*2*2*5

<u>Common factors of the two numbers: </u>

  • 2, 4, 10, 20

Possible number of students as above

4 0
3 years ago
Question is in the picture, please help me out!!
Grace [21]

The arithmetic sequences are as follow:

<h3>What is Arithmetic Sequence?</h3>

An arithmetic sequence in algebra is a sequence of numbers where the difference between every two consecutive terms is the same.

1) t(n) = 5n + 4

t(1) = 9, t(2) = 14, t(3) = 19

So,

9,14,19,...

d= 14-9 = 5

d= 19-14 =5

Hence, it is an AP

2) 1, 2, 4, 8 , 16

Hence, it is not an AP

3) 3, 6, 9 ,...

It is an AP

4)It is given that it is an AP

5) tn =  2*3^n

 t1= 6, t2= 18, t3= 54

So, 6, 18, 54,...

Hence, it is not an AP

6) 3 , 1, 1/3,...

It is not an AP

7) t(n+1)= 6*t(n)

t(1) = -1

t(2)=-6

t(3)= -36

Hence, it is not an AP

8) -3, 1, 5, 9

Hence, it is an AP.

9) 1, 4, 9,...

Hence, it not an AP

10) 2,1,0,1,2,...

It is not an AP

11) t(n)= -2n-5

t(1)= -7, t(2)= -9, t(3)= -11

Hence, it is an AP

12) tn= (1/2)^n

 t1= 1/2, t2= 1/4, t3= 1/8

It is not an AP

Learn more about AP here:

brainly.com/question/24873057

#SPJ1

6 0
2 years ago
The figure shows the dimensions of a wall having a door and window of a room. Write an algebraic expression for the area of the
qwelly [4]

Answer:

Area of the wall to be painted = (11x² + 12x) square units

Step-by-step explanation:

The figure that should be attached to this question is missing. The figure was obtained and is attached to this solution provided.

From the image attached, it is given that the dimension of the rectangular wall to be painted is (4x+3) by (4x), the dimensions of the window is (2x) by (x) and the dimensions of the door is (x) by (3x).

Since, the window space and the door space cannot be painted along with the wall, the Area of the rectangular wall that will be painted will be given by the expression

(Total Area of the rectangular wall) - [(Area of window space) + (Area of door space)]

Area of a rectangular figure = Length × Breadth

Total area of rectangular wall = (4x+3) × 4x = (16x² + 12x) square units

Area of window space = (2x) × (x) = (2x²) square units

Area of door space = (x) × (3x) = (3x²) square units

Area of the wall to be painted = (16x² + 12x) - (2x² + 3x²)

= 16x² + 12x - 5x²

= (11x² + 12x) square units

Hope this Helps!!!

6 0
3 years ago
The dimensions of a rectangle are such that its length is 3 in more than its width if the length were doubled and if the width w
emmasim [6.3K]

Length (L): w + 3            ⇒2(w + 3)

width (w): w                    ⇒ w - 1

Area (A) = L x w

         A = (w + 3)(w)

         A = w² + 3w

*******************************************

       A + 176 = 2(w + 3)(w - 1)

(w² + 3w) + 176 = 2(w + 3)(w - 1)

w² + 3w + 176 = 2w² + 4w - 6

        3w + 176 = w² + 4w - 6

                 176 = w² + w - 6          

                   0 = w² + w - 182

                   0 = (w - 13) (w + 14)

0 = w - 13            0 = w + 14

w = 13                  w = -14

Since width cannot be negative, disregard -14

w = 13

Length (L): w + 3   = (13) + 3   = 16

Answer: width = 13 in, length = 16 in

7 0
3 years ago
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