To evaluate the integral \(\int 2(2x - 3)^{\frac{2}{3}} \mathrm{d}x\), we will use the method of substitution. Let's go through the steps in detail. ### Step 1: Choose a Substitution We notice that the expression \(2x - 3\) is inside the power function. A good substitution would be: \[ u = 2x - 3 \] Now, we need to find \(\mathrm{d}u\): \[ \mathrm{d}u = 2 \mathrm{d}x \quad \Rightarrow \quad \mathrm{d}x = \frac{1}{2} \mathrm{d}u \] ### Step 2: Substitute in the Integral Now we can substitute \(u\) and \(\mathrm{d}x\) into the integral: \[ \int 2(2x - 3)^{\frac{2}{3}} \mathrm{d}x = \int 2u^{\frac{2}{3}} \left(\frac{1}{2} \mathrm{d}u\right) \] This simplifies to: \[ \int u^{\frac{2}{3}} \mathrm{d}u \] ### Step 3: Integrate Next, we integrate \(u^{\frac{2}{3}}\): \[ \int u^{\frac{2}{3}} \mathrm{d}u = \frac{u^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} + C = \frac{u^{\frac{5}{3}}}{\frac{5}{3}} + C = \frac{3}{5} u^{\frac{5}{3}} + C \] ### Step 4: Substitute Back Now we substitute back \(u = 2x - 3\): \[ \frac{3}{5} (2x - 3)^{\frac{5}{3}} + C \] ### Final Answer Thus, the evaluated integral is: \[ \int 2(2x - 3)^{\frac{2}{3}} \mathrm{d}x = \frac{3}{5} (2x - 3)^{\frac{5}{3}} + C \] ### Correct Option The correct option is **A: \(\frac{3}{5}(2x-3)^{\frac{5}{3}} + k\)**. ### Explanation of Other Options - **Option B: \(\frac{6}{5}(2x-3)^{\frac{5}{3}} + k\)**: This option is incorrect because it suggests that the coefficient in front of \((2x - 3)^{\frac{5}{3}}\) is \(\frac{6}{5}\), which does not match our integration result. - **Option C: \(2x - 3 + k\)**: This option is incorrect because it does not reflect the power of \((2x - 3)\) raised to \(\frac{5}{3}\) that we obtained during integration. It also lacks the necessary coefficient. - **Option D: \(2(2x - 3) + k\)**: This option is incorrect as it suggests a linear function rather than the correct power function we derived. ### Summary - We used substitution to simplify the integral. - The integral of \(u^{\frac{2}{3}}\) was calculated correctly. - We substituted back to express the answer in terms of \(x\). - The correct answer is \(\frac{3}{5}(2x - 3)^{\frac{5}{3}} + k\). This method of substitution is a common technique in calculus for handling integrals involving polynomial expressions.