py import symbols, integrate, simplify

Define the variables

x, u = symbols('x u')

Define the substitution

u_sub = 2*x + 1 du_dx = 2 # Derivative of u with respect to x

Integral with substitution

expr = (2x + 1)**3 integral_substituted = expr.subs(2x + 1, u) / du_dx

Simplify the integral expression

integral_result = simplify(integral_substituted)

Compute the limits of integration

a = 2 * 1 + 1 # lower limit when x=1 b = 2 * 3 + 1 # upper limit when x=3 u_limits = (a, b)

integral_result,Using the substitution $u = 2x + 1$, the integral

$\int_1^3 (2x+1)^3 \, dx$

becomes

$\int_3^7 \frac{u^3}{2} \, du,$

where the limits of integration change from $x=1$ to $x=3$ to $u=3$ to $u=7$. The resulting form is $\frac{u^3}{2}$.

Questions for Further Understanding:

1. Would you like to calculate the definite integral value for this?
2. Do you want the integral computed step-by-step for better understanding?
3. Are there other substitution techniques you’d like to compare this with?
4. How does the change in limits affect the calculation here?
5. Would you like examples of similar integral substitutions?

Tip: Always double-check the limits after substitution, as they need to align with the new variable for correctness