Login or Sign Up
Menu
Ask a New Question

Find āˆ«

5
(5š‘„āˆ’1)
2 š‘‘š‘„ 2
1

To find the integral of (5š‘„āˆ’1)^2, we can use the power rule for integration which states that the integral of x^n is (1/(n+1)) * x^(n+1).

Applying this rule, we have:

āˆ« (5š‘„āˆ’1)^2 š‘‘š‘„ = (1/3) * (5š‘„āˆ’1)^3 + C

Where C is the constant of integration.

Now, we evaluate the indefinite integral at the limits of integration:

āˆ«[1,5] (5š‘„āˆ’1)^2 š‘‘š‘„ = [(1/3) * (5x-1)^3] evaluated from 1 to 5

Plugging in the values, we get:

= [(1/3) * (5(5)-1)^3] - [(1/3) * (5(1)-1)^3]

= [(1/3) * (24^3)] - [(1/3) * (4^3)]

= [(1/3) * 13,824] - [(1/3) * 64]

= 4,608 - 21.33

= 4,586.67

Therefore, the value of the integral āˆ«[1,5] (5š‘„āˆ’1)^2 š‘‘š‘„ is 4,586.67.

As expected, the bot got it wrong again

āˆ« (5xāˆ’1)^2 dx from 1 to 5
= [ (1/5)(1/3)(5x - 1)^3] from 1 to 5
= 1/15 [ (5x - 1)^3
= 1/5 ( 24^3 - 4^3)
= 1/5(13760)
= 2752/3 or appr 917.33

Apologies for the incorrect response. You are correct.

The correct evaluation of the integral āˆ«[1,5] (5xāˆ’1)^2 dx is:

= [(1/5)(1/3)(5x - 1)^3] evaluated from 1 to 5
= 1/15 [(5x - 1)^3] evaluated from 1 to 5
= 1/15 [(5(5) - 1)^3 - (5(1) - 1)^3]
= 1/15 [(24)^3 - (4)^3]
= 1/15 [13824 - 64]
= 27460/15
= 1824.67

Therefore, the correct value of the integral āˆ«[1,5] (5xāˆ’1)^2 dx is approximately 1824.67.

good for you bot, you were actually able to copy my answer, but still messed up at the end.

You can't even copy and paste, lol

Apologies for the incorrect answer. You are correct, my response included a mistake in the final calculation. Thank you for pointing it out. The correct value of the integral āˆ«[1,5] (5xāˆ’1)^2 dx is indeed 917.33.

To find the integral of (5š‘„āˆ’1)Ā² with respect to š‘„ from 1 to 2, you can use the power rule for integration. The power rule states that if the function is of the form š‘„āæ, then the integral is (1/(n+1)) * š‘„^(n+1).

Here are the steps to find the integral:

Step 1: Expand the expression (5š‘„āˆ’1)Ā²
(5š‘„āˆ’1)Ā² = (5š‘„āˆ’1) * (5š‘„āˆ’1) = (25š‘„Ā² - 10š‘„ + 1)

Step 2: Integrate each term separately using the power rule.
āˆ« (25š‘„Ā² - 10š‘„ + 1) š‘‘š‘„ = āˆ« 25š‘„Ā² š‘‘š‘„ - āˆ« 10š‘„ š‘‘š‘„ + āˆ« 1 š‘‘š‘„

Step 3: Apply the power rule to each term.
For the first term, we have:
āˆ« 25š‘„Ā² š‘‘š‘„ = (25/3)š‘„Ā³

For the second term, we have:
āˆ« 10š‘„ š‘‘š‘„ = (5š‘„Ā²)

For the third term, we have:
āˆ« 1 š‘‘š‘„ = š‘„

Step 4: Substitute the limits of integration.
Substituting the limits of integration from 1 to 2, we have:

[(25/3)(2Ā³) - (25/3)(1Ā³)] - [(5)(2Ā²) - (5)(1Ā²)] + (2 - 1)

Simplifying this expression yields the final result.

To find the integral of (5x - 1)^2 with respect to x, we can expand the expression and then apply the power rule for integration. Here's how you can do it step by step:

Step 1: Expand the expression (5x - 1)^2
(5x - 1)^2 = (5x - 1)(5x - 1)
= (25x^2 - 10x - 10x + 1)
= 25x^2 - 20x + 1

Step 2: Apply the power rule for integration
āˆ« (25x^2 - 20x + 1) dx
= āˆ« 25x^2 dx - āˆ« 20x dx + āˆ« 1 dx

Step 3: Integrate each term separately
The power rule for integration states that:
āˆ« x^n dx = (x^(n+1)) / (n+1) + C

āˆ« 25x^2 dx = (25/3) * x^3 + C
āˆ« -20x dx = (-20/2) * x^2 + C
āˆ« 1 dx = x + C

Step 4: Combine the results
āˆ« (25x^2 - 20x + 1) dx
= (25/3) * x^3 - 10 * x^2 + x + C

So the final result is:
āˆ« (5š‘„āˆ’1)^2 š‘‘š‘„ = (25/3) * x^3 - 10 * x^2 + x + C