Evaluate the following indefinite integrals.
∫4e^6x dx
∫radical8x^8 raised to the third
power dx
There is nothing to evaluate. If copy and paste did not work, please repost with the needed data typed. Thanks for asking.
To evaluate the indefinite integrals, let's take them one by one:
1) ∫4e^6x dx:
To evaluate this integral, we can use the power rule of integration. The power rule states that If f(x) = ax^n, then ∫f(x) dx = (a/(n+1)) * x^(n+1) + C, where C represents the constant of integration.
In the given integral, f(x) = 4e^6x, where a = 4 and n = 1 (since e^6x can be written as (e^6)^x, and (e^6) is a constant, making x the variable with power 1).
So, by using the power rule, we have:
∫4e^6x dx = (4/(1+1)) * (e^6x)^(1+1) + C
= 2 * (e^6x)^2 + C
= 2 * e^(12x) + C
Therefore, the indefinite integral of 4e^6x dx is 2e^(12x) + C, where C is the constant of integration.
2) ∫√(8x^8)^3 dx:
To evaluate this integral, let's simplify the expression first.
We have √(8x^8)^3 = √(8^3) * (x^8)^3 = 2^3 * x^(8 * 3) = 8x^24.
Now, we can use the power rule of integration to evaluate the integral.
∫(8x^24) dx = (8/(24+1)) * x^(24+1) + C
= (8/25) * x^25 + C
Therefore, the indefinite integral of √(8x^8)^3 dx is (8/25) * x^25 + C, where C is the constant of integration.