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.
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Here are the solutions to the integration problems: (i) To evaluate (( x))/(x) dx: Step 1: Let u = x. Then, differentiate u with respect to x: du = (1)/(x) dx Step 2: Substitute u and du into the integral. (u) du Step 3: Integrate with respect to u. -(u) + C Step 4: Substitute back u = x. -( x) + C (ii) To evaluate ((3 x + 4 x)/(4 x - 3 x)) dx: Step 1: Let u = 4 x - 3 x. Then, differentiate u with respect to x: du = (4 x - 3(- x)) dx du = (4 x + 3 x) dx Step 2: Substitute u and du into the integral. (1)/(u) du Step 3: Integrate with respect to u. |u| + C Step 4: Substitute back u = 4 x - 3 x. |4 x - 3 x| + C (iii) To evaluate e^m ^-1 x1+x^2 dx: Step 1: Let u = m ^-1 x. Then, differentiate u with respect to x: du = m · (1)/(1+x^2) dx Rearrange to find (1)/(1+x^2) dx: (1)/(1+x^2) dx = (1)/(m) du Step 2: Substitute u and du into the integral. e^u · (1)/(m) du (1)/(m) e^u du Step 3: Integrate with respect to u. (1)/(m) e^u + C Step 4: Substitute back u = m ^-1 x. (1)/(m) e^m ^-1 x + C (iv) To evaluate x (4x^2 + 7) dx: Step 1: Let u = 4x^2 + 7. Then, differentiate u with respect to x: du = 8x \, dx Rearrange to find x \, dx: x \, dx = (1)/(8) du Step 2: Substitute u and du into the integral. (u) · (1)/(8) du (1)/(8) (u) du Step 3: Integrate with respect to u. (1)/(8) (-(u)) + C -(1)/(8) (u) + C Step 4: Substitute back u = 4x^2 + 7. -(1)/(8) (4x^2 + 7) + C Drop the next question! 📸