Integrate (x+3)^(1/2)dx

1. The task at hand involves the integration of the expression (x + 3) raised to the power of one-half with respect to the variable x. This expression can be interpreted as the square root of the quantity (x + 3). To approach this integration, one might consider employing a substitution method, which can simplify the process and make it more manageable. By letting u equal (x + 3), the differential dx can be expressed in terms of du, thereby transforming the integral into a more straightforward form.

2. Upon making the substitution u = (x + 3), the differential dx becomes du, and the limits of integration will adjust accordingly if definite limits are provided. The integral now takes the form of the square root of u, specifically u^(1/2). This new expression can be integrated using the power rule for integration, which states that the integral of u^n is (u^(n + 1))/(n + 1) plus a constant of integration, provided that n is not equal to -1. In this case, the exponent is one-half, allowing for a direct application of the power rule.

3. After performing the integration, the result will yield (2/3)u^(3/2) plus the constant of integration C. Substituting back the original variable, the final expression for the integral of (x + 3)^(1/2)dx becomes (2/3)(x + 3)^(3/2) + C. This result encapsulates the area under the curve represented by the function (x + 3)^(1/2) with respect to x, providing a comprehensive solution to the integration problem presented.