Find the most general antiderivative of the function
1) f(x)= (X^3)^1/4 + (X^4)^1/3
2) f(u)= U^4 + 3�ãU / U^2
To find the most general antiderivative of a function, you need to apply integration rules and techniques. Let's solve each problem step by step:
1) f(x) = (x^3)^(1/4) + (x^4)^(1/3)
To integrate each term, we can use the power rule of integration:
∫ x^n dx = (1/(n+1)) * x^(n+1)
For the first term, (x^3)^(1/4), we can rewrite it as x^(3/4). Applying the power rule, we get:
∫ x^(3/4) dx = (1/(3/4 + 1)) * x^(3/4 + 1) = (4/7) * x^(7/4)
For the second term, (x^4)^(1/3), we can rewrite it as x^(4/3). Applying the power rule, we get:
∫ x^(4/3) dx = (1/(4/3 + 1)) * x^(4/3 + 1) = (3/7) * x^(7/3)
Putting it all together, the most general antiderivative of the function f(x) is:
F(x) = (4/7) * x^(7/4) + (3/7) * x^(7/3) + C,
where C is the constant of integration.
2) f(u) = u^4 + 3√u / u^2
To integrate the first term, u^4, we can use the power rule as before:
∫ u^4 du = (1/(4 + 1)) * u^(4 + 1) = (1/5) * u^5
For the second term, 3√u, we can rewrite it using fractional exponents:
3√u = u^(1/3)
Applying the power rule, we get:
∫ u^(1/3) du = (1/(1/3 + 1)) * u^(1/3 + 1) = (3/4) * u^(4/3)
For the third term, u^2, we can use the power rule again:
∫ u^2 du = (1/(2 + 1)) * u^(2 + 1) = (1/3) * u^3
Putting it all together, the most general antiderivative of the function f(u) is:
F(u) = (1/5) * u^5 + (3/4) * u^(4/3) + (1/3) * u^3 + C,
where C is the constant of integration.