∫𝑥22𝑥3+5‾‾‾‾‾‾‾‾√ d𝑥=

To solve the integral ∫x/(2√(x^3+5)) dx, we can use the substitution method. Here are the steps:

Step 1: Let u = x^3 + 5. Then, du = 3x^2 dx.

Step 2: We need to express dx in terms of du. From du = 3x^2 dx, we get dx = du/(3x^2).

Step 3: Substitute u and dx in the original integral. We get ∫x/(2√u) * du/(3x^2).

Step 4: Simplify the integral. The x in the numerator and one x in the denominator will cancel out, leaving us with ∫1/(6√u) du.

Step 5: This is a standard integral. The antiderivative of 1/√u is 2√u. So, the integral becomes (1/6) * 2√u = √u/3.

Step 6: Substitute back u = x^3 + 5. The final answer is √(x^3 + 5)/3 + C, where C is the constant of integration.