if integral of 2 to -2 (x^7+k)dx=16, then k=
a)-12
b)12
c)-4
d)4
e)0
Nice
Well, solving an integral equation requires some number crunching, but solving it with humor? Challenge accepted!
Let's put on our mathematical clown noses and see what we can do!
First, we need to find the antiderivative of (x^7 + k). The antiderivative of x^n (where n is a constant) is (x^(n+1))/(n+1).
So, the antiderivative of (x^7 + k) is [(x^8)/8 + kx] + C, where C represents the constant of integration.
Next, we evaluate the definite integral from 2 to -2:
[(2^8)/8 + k(2)] - [( (-2)^8)/8 + k(-2)] = 16
Wait a minute, we're getting serious here. Let's add some humor back into the mix!
Now, it's time to simplify this expression and find the value of k. We have:
[(256)/8 + 2k] - [(256)/8 - 2k] = 16
With a little algebraic juggling, the 2k terms cancel out, leaving:
256/8 - 256/8 = 16
Since 0 = 16 is clearly not true, let's conclude that there must be a typo!
But hey, there's always a silver lining with humor! So, here's a funny answer for you:
The value of k is hidden in the wonderful land of typos, and only the mischievous clowns of the circus can find it!
But in all seriousness, it seems like there is no real value of k that satisfies the equation. So, the answer is e) 0.
To find the value of k, we can solve the definite integral first.
Given that the integral of (x^7 + k)dx from 2 to -2 equals 16, we can write the equation as follows:
∫[2 to -2] (x^7 + k) dx = 16
To solve this integral, we can split it into two separate integrals:
∫[2 to -2] x^7 dx + ∫[2 to -2] k dx = 16
Now let's evaluate each integral step-by-step:
1) ∫[2 to -2] x^7 dx:
To evaluate this integral, we can use the power rule of integration, which states that if we have an integral of the form ∫ x^n dx, then the result is (1/(n+1)) * x^(n+1).
Applying this rule, we get:
(1/8) * x^8 from 2 to -2
Now substituting the upper and lower limits into the equation:
(1/8) * (-2)^8 - (1/8) * (2)^8
Simplifying further:
(1/8) * 256 - (1/8) * 256 = 0
2) ∫[2 to -2] k dx:
Since k is just a constant, we can treat it as k times the integral of dx, which is just x.
Therefore, ∫[2 to -2] k dx = k * x from 2 to -2
Substituting the upper and lower limits:
k * (-2) - k * 2
Simplifying further:
-2k - 2k = -4k
Now, combining both results, we get:
0 + (-4k) = 16
-4k = 16
Dividing both sides by -4:
k = -4
Therefore, the value of k is -4, which corresponds to the option (c) in the given choices.
To find the value of k in the integral, we can use the fundamental theorem of calculus. According to the theorem, the integral of a function f(x) from a to b is given by F(b) - F(a), where F(x) is the antiderivative of f(x).
In this case, we have the integral from 2 to -2 of (x^7 + k) dx. Let's find the antiderivative of (x^7 + k) with respect to x. The antiderivative of x^7 is (1/8)x^8, and the antiderivative of a constant k is just kx. So, the antiderivative of (x^7 + k) is (1/8)x^8 + kx.
Now, we can evaluate the integral using the antiderivative. The integral from 2 to -2 of (x^7 + k) dx is equal to [(1/8)(-2)^8 + k(-2)] - [(1/8)(2)^8 + k(2)].
Simplifying this expression, we get [(1/8)(256) - 2k] - [(1/8)(256) + 2k].
Combining like terms, the expression becomes (32 - 2k) - (32 + 2k).
Distributing the negative sign, we have 32 - 2k - 32 - 2k.
Simplifying further, we get -4k.
We are given that the value of the integral is 16, so we can set -4k equal to 16 and solve for k:
-4k = 16
Dividing both sides by -4:
k = -4.
Therefore, the value of k is -4, which corresponds to option c) -4.
The x^7 part of the integral is zero because it is an "odd" function, so the 2 to -0 and 0 to -2 parts cancel out.
Integral of k dx = -4 k .
2 -> -2
Therefore -4k = 16 and
k = -4