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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Q10. Evaluate the integral: ((x)/(x+1)) dx Step 1: Rewrite the integrand using algebraic manipulation. (x)/(x+1) = (x+1-1)/(x+1) = 1 - (1)/(x+1) Step 2: Integrate each term. (1 - (1)/(x+1)) dx = 1 \, dx - (1)/(x+1) \, dx Step 3: Perform the integration. 1 \, dx = x (1)/(x+1) \, dx = |x+1| Step 4: Combine the results and add the constant of integration. x - |x+1| + C The final answer is x - |x+1| + C. Q11. Evaluate the integral: ^2 x \, dx Step 1: Use the trigonometric identity ^2 x = ^2 x - 1. ^2 x \, dx = (^2 x - 1) \, dx Step 2: Integrate each term. ^2 x \, dx - 1 \, dx Step 3: Perform the integration. ^2 x \, dx = x 1 \, dx = x Step 4: Combine the results and add the constant of integration. x - x + C The final answer is x - x + C. Q12. Compute the indefinite integral: sqrt(1- 2x) \, dx Step 1: Use the identities 1 = ^2 x + ^2 x and 2x = 2 x x to rewrite the expression under the square root. 1 - 2x = ^2 x + ^2 x - 2 x x = ( x - x)^2 Step 2: Substitute this back into the integral. sqrt(( x - x)^2) \, dx = | x - x| \, dx Step 3: For simplicity, assume x - x 0 over the interval of integration, allowing us to remove the absolute value. ( x - x) \, dx Step 4: Integrate each term. x \, dx - x \, dx Step 5: Perform the integration. x \, dx = x x \, dx = - x Step 6: Combine the results and add the constant of integration. x - (- x) + C = x + x + C The final answer is x + x + C. Q13. Compute the integral: (^2 x)/(1+ x) \, dx Step 1: Use the trigonometric identity ^2 x = 1 - ^2 x. (1 - ^2 x)/(1+ x) \, dx Step 2: Factor the numerator using the difference of squares formula a^2 - b^2 = (a-b)(a+b). 1 - ^2 x = (1 - x)(1 + x) Step 3: Substitute the factored numerator back into the integral and simplify. ((1 - x)(1 + x))/(1+ x) \, dx = (1 - x) \, dx Step 4: Integrate each term. 1 \, dx - x \, dx Step 5: Perform the integration. 1 \, dx = x x \, dx = x Step 6: Combine the results and add the constant of integration. x - x + C The final answer is x - x + C. Send me the next one 📸