Applications

Suppose that f(x) = d/dx (1 − √x...

Question

Applications

Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).

Find:

∫[f(x) + g(x)] dx

Accepted Answer

Welcome back everyone. In this problem, let f x be the derivative of 3 minus x to the power of 3 halves, and the g x be the derivative of 2 x minus 5. Find the integral of fx plus g x with respect to x. Now what can we do that will help us to find the integral? Well, we know that the integral of the sum of fx and the gfx with respect to x. It is basically the same thing as the integral of fx with respect to x plus the integral of g of x with respect to x. So if we can figure out both of these integrals, then we should be able to solve our problem. So how can we do that? Well, recall that if the derivative of the antiderivative of F of X is equal to FFX, OK, then the integral of FX with respect to x is going to be equal to the antiderivative of FX plus a constant k1. So what are we saying? Well, basically, if we can express our function FX in terms of our anti derivative, if we can integrate, then we should be able to solve. And here we know what we need to do to integrate. So in this case. For example, if we let the antiderivative of fx be equal to 3 minus x to the power of 3 halves, OK, then. The integral of fx with respect to x is going to be equal to the ante derivative of fx plus our constant k1, which is going to be 3 minus x to the power of 3 halves plus k1, OK, because remember, as we said. If we were to integrate the derivative of 3 minus x to the power of 3 halves, we would end up with 3 minus x 3 halves, since integration is the opposite of differentiation. Likewise, If we let GFX, applying the same idea, the the antiderivative of GFX be equal to 2 x minus 5, OK, then we can say. That the derivative of geoic. With respect to x will be equal to the ante derivative of g of x plus some constant k2, and that's going to be 2 x minus 5 plus k2. So now that we have both of the integrals of FX and GFX, we can apply it to our problem because no, this means then. That the integral of FX plus GX with respect to x is going to be equal to 3 minus x the power of 3 halves plus K1. + 2 x minus 5 + k2 If we simplify that a bit further. Then we're going to have 2 eggs. Minus x to the power of 3 halves plus 3 minus 5, which is -2 + k1 + K2. Now we can take this even further because K 2 + K1 + K2 is our constants, so we can combine those into a single integration constant C. Therefore, the result or or integral is going to be 2 x minus x to the power of 3 halves plus some constant c. Thanks a lot for watching everyone. I hope this video helped.

ApplicationsSuppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).Find:∫[f(x) + g(x)] dx

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Applications

Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).

Find:

∫[f(x) + g(x)] dx