I have to find the integral of
(e^3lnx + e^3x)dx. Can someone help me?
Would the answer be (x^4)/4 + (e^3x)/3 + C?
Is that (e^3)(lnx) + (e^3)(x) or
e^(3lnx) + e^(3x)?
Assuming the latter, since it's slightly more interesting than e^3*(lnx + x)
e^(3lnx)dx = x^3 dx
e^(3x)dx = 1/3 e^u du where u = 3x
That should be easy
Well, well, well, looks like we have an integral to solve!
Let's break it down step by step. First, we need to find the integral of each term separately.
Starting with e^3lnx, we can use a little trick called logarithmic differentiation. We rewrite it as x^3 and apply the power rule for integration, which gives us (1/4)x^4.
Now, onto the second term, e^3x. This is a simple exponential function, so we can integrate it by using the power rule again. The integral of e^ax is simply (1/a)e^ax. Applying this, we get (1/3)e^3x.
Finally, we add the results together: (1/4)x^4 + (1/3)e^3x.
So, the integral of (e^3lnx + e^3x)dx is (1/4)x^4 + (1/3)e^3x + C.
Voila! You've now integrated your way to mathematical success! Keep up the good work!
Of course! I can help you find the integral of (e^3lnx + e^3x)dx. Let's tackle each term separately.
1. Integral of e^3lnx dx:
To integrate e^3lnx, we can use the property that e^lnx = x. So, let's replace e^3lnx with x^3.
∫ e^3lnx dx = ∫ x^3 dx
2. Integral of e^3x dx:
To integrate e^3x, we can use the power rule for integrals. The integral of e^ax dx is (1/a) * e^ax. In this case, a = 3.
∫ e^3x dx = (1/3) * e^3x
Now, let's put the two integrals together.
∫ (e^3lnx + e^3x) dx = ∫ x^3 dx + ∫ (1/3) * e^3x dx.
Now, let's integrate each term:
∫ x^3 dx = (1/4) * x^4 + C1, where C1 is the constant of integration.
∫ (1/3) * e^3x dx = (1/3) * (1/3) * e^3x + C2, where C2 is another constant of integration.
Finally, combining the two integrals:
∫ (e^3lnx + e^3x) dx = (1/4) * x^4 + (1/3) * (1/3) * e^3x + C1 + C2.
So, the final answer is (1/4) * x^4 + (1/9) * e^3x + C, where C = C1 + C2 is the combined constant of integration.
Sure, I can help you with that. To find the integral of (e^3lnx + e^3x)dx, you can use the linearity property of integration.
First, let's consider the integral of e^3lnx. We can simplify this expression using the logarithmic rule for exponentiation, which states that e^lnx = x. Therefore, e^3lnx can be rewritten as (e^lnx)^3 = x^3.
Now, let's compute the integral of x^3. To integrate a power of x, you can use the power rule of integration. The power rule states that the integral of x^n dx, where n is any real number except -1, is (x^(n+1))/(n+1). Applying the power rule to x^3, we get (1/4)x^4.
Next, let's consider the integral of e^3x. This is a standard integral of an exponential function. The integral of e^nx dx is simply (1/n)e^nx. Therefore, the integral of e^3x is (1/3)e^3x.
Now, we can put it all together. The integral of (e^3lnx + e^3x)dx is equal to the integral of x^3 + e^3x dx, which is ((1/4)x^4 + (1/3)e^3x) + C, where C is the constant of integration.
So, the final result is ((1/4)x^4 + (1/3)e^3x) + C.