Not sure how to find the answer

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Verizon [17]

If f(x) has an inverse on [a, b], then integrating by parts (take u = f(x) and dv = dx), we can show

$\displaystyle \int_a^b f(x) \, dx = b f(b) - a f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) \, dx$

Let $f(x) = \sqrt{1 + x^3}$ . Compute the inverse:

$f\left(f^{-1}(x)\right) = \sqrt{1 + f^{-1}(x)^3} = x \implies f^{-1}(x) = \sqrt[3]{x^2-1}$

and we immediately notice that $f^{-1}(x+1)=\sqrt[3]{x^2+2x}$ .

So, we can write the given integral as

$\displaystyle \int_0^2 f^{-1}(x+1) + f(x) \, dx$

Splitting up terms and replacing $x \to x-1$ in the first integral, we get

$\displaystyle \int_1^3 f^{-1}(x) \, dx + \int_0^2 f(x) \, dx = 2 f(2) - 0 f(0) = 2\times3+0\times1=\boxed{6}$

7 0

1 year ago

Evaluate 4|-2| - |-3|

Nitella [24]

As we know these two lines mean | | a modulus sign. The modulus symbol is sometimes used in conjunction with inequalities. A modulus sign in simple words just ignores the sign and gives us the absolute value.

For example in this case the modulus gives us 4 x 2 - 3. As you can see the modulus sign has ignored the original negative signs with the numbers.

So after evaluating the equation the answer comes out to be 8 - 3 which is equal to 5.

5 0

3 years ago

The blue lake trail is 11 3/8 miles long. Gemma has hiked 2 1/2 miles each hour for 3 hours. How far is she from the end of the

melomori [17]

In the question it is given that the length of the blue lake trail is 11 3/8 miles. In the question it is also given that Gemma has hiked for a total time of 3 hours and each hour she has hiked 2 1/2 miles.
Now coming to solving the problem we get:
Total length of blue lake trail = 11 3/8 miles
   = 91/8 miles
Distance hiked by Gemma in 1 hour = 2 1/2 miles
   = 5/2 miles
Then the total distance hiked by Gemma in 3 hours = (5/2) * 3
   = 15/2 miles
The distance left to cover the full trail = (91/8) - (15/2) miles
      = (91 - 60)/8 miles
   = 31/8 miles
   = 3 7/8 miles.
So Gemma is 31/8 miles or 3 7/8 miles from the end of the trail.

7 0

3 years ago

(10,6) and (2,4) The midpoint of the segment is

Sholpan [36]

The midpoint is the segment is 6,5

7 0

3 years ago

Read 2 more answers

Marjorie bought 24 bottles of juice. Each day she opens and drinks 2 of these bottles of juice. Which inequality represents the

Anika [276]

Answer:

2d≤24

Step-by-step explanation:

Total number of bottled juice bought = 24 bottles

Amount she drank each day = 2 bottles

To get the inequality to represent greatest number of days she can continue before she has to replenish her stock, we will say;

1 day = 2 bottles

d days = 24 bottles

Cross multiply

2 * d = 1 * 24

2d = 24

Since we want to know the greatest number of days she can continue before she has to replenish her stock, the equal to to sign will be replaced with less than or equal to sign as shown;

2d≤24

This shows that the maximum number of days she can continue before replenishing is 12days. It cannot exceed

6 0

3 years ago