i need help with finding
anti derivative of
-2xe^x +y-6x^2 dx
so i use the integration by parts to solve for -2xe^x?
Yes. Let x = u and dv = e^x dx
du = 1 and v = e^x
So the integral of x e^x is
u v - integral of v du
= x e^x - e^x
what is the antiderivitive of the squre root of two
Yes, you can use integration by parts to find the antiderivative of the term -2xe^x in the given expression. The integration by parts formula states:
∫ u * dv = u * v - ∫ v * du
To apply the formula, we need to choose u and dv. Let's assign:
u = -2x (derivative of u with respect to x is du = -2 dx)
dv = e^x dx (integral of dv with respect to x is v = ∫ e^x dx = e^x)
Now, we can calculate the values of u, du, v, and ∫v du:
u = -2x
du = -2 dx
v = e^x
∫ v du = ∫ e^x (-2 dx)
Using the integration by parts formula, we have:
∫ u dv = u * v - ∫ v du
Applying the formula, we get:
∫ -2xe^x dx = -2x * e^x - ∫ e^x (-2 dx)
Simplifying, we have:
∫ -2xe^x dx = -2x * e^x + 2 ∫ e^x dx
The second term in the above expression, ∫ e^x dx, is the integral of e^x, which is simply e^x. So, we have:
∫ -2xe^x dx = -2x * e^x + 2 * e^x + C
where C represents the constant of integration.
Thus, the antiderivative of -2xe^x is -2x * e^x + 2 * e^x + C.