Consider the function f(x)=x^4+2sqrt x. Let F(x) be the anti-derivative of f(x) with F(1)=-9.
Then F(x)= ?
oops.
F(x) = 1/5 x^5 + 2 * 2/3 x^(3/2) + C
1/5 * 1 + 2 * 2/3 * 1 + C = -9
This is just like your other problem.
f(x) = x^4 + 2√x
F(x) = 1/5 x^5 + 2/3 x^(3/2) + C
Now use F(1) = -9 to find C
1/5 * 1 + 2/3 * 1 + C = -9
and finish it off.
F(x)=-10/x+2/5x^5+48/5
To find the anti-derivative F(x) of the given function f(x), we need to integrate f(x) with respect to x.
Given f(x) = x^4 + 2sqrt(x)
To integrate x^4, we add 1 to the exponent and divide by the new exponent:
∫ x^4 dx = (1/5) x^5 + C
To integrate 2sqrt(x), we use the power rule for integration.
∫ sqrt(x) dx = (2/3) x^(3/2) + C
Now, adding the two integrals together, we have:
F(x) = (1/5) x^5 + (2/3) x^(3/2) + C
We are given that F(1) = -9. Substituting x = 1 into the equation, we get:
-9 = (1/5) + (2/3) + C
To find the value of C, we can solve the equation:
C = -9 - (1/5) - (2/3)
C = -9 - (3/15) - (10/15)
C = -9 - (13/15)
C = -144/15 - 13/15
C = -157/15
Finally, the anti-derivative F(x) with F(1) = -9 is:
F(x) = (1/5) x^5 + (2/3) x^(3/2) - (157/15)