This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To evaluate the definite integral _0^1 x^2 (x^3 + 2)^3 \,dx, we will use the method of substitution. Step 1: Define the substitution. Let u = x^3 + 2. Step 2: Find the differential du. Differentiate u with respect to x: (du)/(dx) = (d)/(dx)(x^3 + 2) = 3x^2 Rearrange to find dx in terms of du: du = 3x^2 \,dx x^2 \,dx = (1)/(3) du Step 3: Change the limits of integration. Since the integral is definite, we need to change the limits from x-values to u-values. When x = 0: u = (0)^3 + 2 = 0 + 2 = 2 When x = 1: u = (1)^3 + 2 = 1 + 2 = 3 So the new limits of integration are from u=2 to u=3. Step 4: Rewrite the integral in terms of u. Substitute u and x^2 \,dx into the integral: _0^1 x^2 (x^3 + 2)^3 \,dx = _2^3 u^3 ((1)/(3) du) = (1)/(3) _2^3 u^3 \,du Step 5: Integrate with respect to u. Apply the power rule for integration, u^n \,du = u^n+1n+1: (1)/(3) [ u^3+13+1 ]_2^3 = (1)/(3) [ (u^4)/(4) ]_2^3 Step 6: Evaluate the definite integral using the new limits. Substitute the upper limit and subtract the result of substituting the lower limit: = (1)/(3) ( ((3)^4)/(4) - ((2)^4)/(4) ) = (1)/(3) ( (81)/(4) - (16)/(4) ) = (1)/(3) ( (81 - 16)/(4) ) = (1)/(3) ( (65)/(4) ) = (65)/(12) The final answer is (65)/(12)