Mathematics : asked on lasardia
 09.04.2023

The indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent. 6x^5(x^6-2)dx

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24.06.2023, solved by verified expert
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The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

Step-by-step explanation:

To find:

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

Solution:

Method of substitution:

Let The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

Differentiate both sides with respect to The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

[use The indefinite integral can be found in more, №17886891, 09.04.2023 20:39]

So,

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = ∫ The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 where The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 is a variable.

(Use ∫The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 )

Put The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

Use The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

So,

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

where The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

Without using substitution:

The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = ∫The indefinite integral can be found in more, №17886891, 09.04.2023 20:39 = The indefinite integral can be found in more, №17886891, 09.04.2023 20:39

So, same answer is obtained in both the cases.

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Mathematics
Step-by-step answer
P Answered by PhD

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.

Mathematics
Step-by-step answer
P Answered by PhD
Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram
Mathematics
Step-by-step answer
P Answered by PhD

Cost of 7 gallons=$24.50

Cost of 1 gallon=24.50/7=3.5

Cost of 15 gallons=15*3.5=52.5

Cost of 15 gallons will be $52.5

Mathematics
Step-by-step answer
P Answered by PhD

The answer is in the image 

The answer is in the image 
Mathematics
Step-by-step answer
P Answered by PhD

F=ma

where F=force

m=mass

a=acceleration

Here,

F=4300

a=3.3m/s2

m=F/a

    =4300/3.3

    =1303.03kg

Approximately it is aqual to 1300kg

Mathematics
Step-by-step answer
P Answered by PhD

Salesperson will make 6% of 1800

=(6/100)*1800

=108

Salesperson will make $108 in $1800 sales

Mathematics
Step-by-step answer
P Answered by PhD

Speed=Distance/time

Here,

distance=15m

time=1sec

speed=15/1=15m/sec

Distance=Speed*time

time=15min=15*60sec=900sec

Distance travelled in 15 min=15*900=13,500m

=13500/1000 km=13.5Km

Mathematics
Step-by-step answer
P Answered by PhD

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

Mathematics
Step-by-step answer
P Answered by PhD

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

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