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Darina [25.2K]
2 years ago
14

Integrate the question below​

Mathematics
1 answer:
Natalka [10]2 years ago
8 0

It looks like the integral is

\displaystyle \int_0^1 \frac{2x - 8x^2}{1+4x} \, dx

Polynomial division yields

\dfrac{2x-8x^2}{1+4x} = -2x + 1 - \dfrac1{1+4x}

Now split up the integral as

\displaystyle \int_0^1 \left(-2x + 1 - \frac1{1+4x}\right) \, dx = \int_0^1 (1-2x) \, dx - \int_0^1 \frac{dx}{1+4x}

In the second integral, substitute u=1+4x and du=4\,dx. Then x=0 \implies u=1 and x=1 \implies u=5, so

\displaystyle \int_0^1 \frac{dx}{1+4x} = \frac14 \int_1^5 \frac{du}u

and by the fundamental theorem of calculus, the integral we want evaluates to

\displaystyle \int_0^1 \left(-2x + 1 - \frac1{1+4x}\right) \, dx = \int_0^1 (1-2x) \, dx - \frac14 \int_1^5 \frac{du}u \\\\ = (x - x^2)\bigg|_{x=0}^{x=1} - \frac14 \ln|u| \bigg|_{u=1}^{u=5} \\\\ = \bigg((1 - 1^2) - (0 - 0^2)\bigg) - \frac14 (\ln(5) - \ln(1)) = \boxed{-\frac{\ln(5)}4}

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Select the correct answer.
adell [148]

Answer:

its c. x = -2.5 and x = -2

Step-by-step explanation:

multiply function (g)(x) = (-2x-5)(x-2)

= -2x^2+4x-5x+10

first get out -2x(x+2)-5(x+2)

(-2x-5) or (x+2) = 0

x=-5/2 or x=-2

its c that both points covered in that line of x axis

8 0
2 years ago
Does anyone know how to do this? I know it's easy but I forgot please help
kenny6666 [7]

Answer: 4

Step-by-step exzplanation: 20/360= X/72

360X= 1440

Divide by 360

X=4

5 0
3 years ago
Triangle ABC has vertices A(0,6) B(4,6) C(1,3). Sketch a graph of ABC and use it to find the orthocenter of ABC. Then list the s
Allisa [31]

For line B to AC:  y - 6 = (1/3)(x - 4);  y - 6 = (x/3) - (4/3); 3y - 18 = x - 4, so 3y - x = 14

For line A to BC:  y - 6 = (-1)(x - 0);  y - 6 = -x, so y + x = 6

Since these lines intersect at one point (the orthocenter), we can use simultaneous equations to solve for x and/or y:

(3y - x = 14) + (y + x = 6) =>  4y = 20, y = +5;  Substitute this into y + x = 6:  5 + x = 6, x = +1

<span>So the orthocenter is at coordinates (1,5), and the slopes of all three orthocenter lines are above.</span>

5 0
3 years ago
What is the equation of the circle with center (−3, 1) that passes through the point (−5, 3)?
Mariulka [41]

Answer:

( x + 3)^2 + ( y - 1)^2 = 8

Step-by-step explanation:

The equation of a circle with a center and a point

( x - a) ^2 + ( y - b) ^2 = r^2

( x , y) - any point on the circle

( a, b) - center of the circle

r^2 - radius of the circle

( -3 , 1) - center - ( a, b)

a = -3

b = 1

( -5 , 3) - point - ( x, y)

x = -5

y = 3

Step 1: substitute the center into the equation

( x -(-3)^2 + ( y - 1)^2 = r^2

( x + 3)^2 + ( y - 1)^2 = r^2

Step 2 : sub the point into the equation

( x + 3)^2 + ( y - 1)^2 = r^2

( -5 + 3)^2 + ( 3 - 1)^2 = r^2

(-2)^2 + 2^2 = r^2

4 + 4 = r^2

8 = r^2

Step 3 : sub the radius into the equation

( x + 3)^2 + ( y - 1)^2 = r^2

( x + 3)^2 + (y - 1)^2 = 8

Therefore, the equation of the circle is

( x + 3)^2 + (y - 1)^2 = 8

6 0
3 years ago
Question in photo bc i cant type it right lol
Gwar [14]
It would be a i hope this helps 
6 0
3 years ago
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