Solve the following indefinite integrals.
Answer (1)
Answer:2. ∫(3x + 4)² dx = 3x³ + 12x² + 16x + C3. ∫x(√x − 1) dx = (2/5)x^(5/2) − (1/2)x² + C4. ∫(x + 4)/√x dx = (2/3)x^(3/2) + 8x^(1/2) + C5. ∫(4/x³ + 2/x) dx = −2x^(-2) + 2ln|x| + CStep-by-step explanation:2. ∫(3x + 4)² dxFirst expand:(3x + 4)² = 9x² + 24x + 16Now integrate:∫(9x² + 24x + 16) dx = 3x³ + 12x² + 16x + CAnswer: 3x³ + 12x² + 16x + C--------------------------------------------------3. ∫x(√x − 1) dxDistribute x:x(√x − 1) = x^(3/2) − xNow integrate:∫(x^(3/2) − x) dx = (2/5)x^(5/2) − (1/2)x² + CAnswer: (2/5)x^(5/2) − (1/2)x² + C--------------------------------------------------4. ∫(x + 4)/√x dxSplit the terms:(x + 4)/√x = x/√x + 4/√x = x^(1/2) + 4x^(-1/2)Now integrate:∫(x^(1/2) + 4x^(-1/2)) dx = (2/3)x^(3/2) + 8x^(1/2) + CAnswer: (2/3)x^(3/2) + 8x^(1/2) + C--------------------------------------------------5. ∫(4/x³ + 2/x) dxRewrite as powers:4/x³ + 2/x = 4x^(-3) + 2x^(-1)Now integrate:∫(4x^(-3) + 2x^(-1)) dx = 4 * (-1/2)x^(-2) + 2ln|x| + C= -2x^(-2) + 2ln|x| + CAnswer: -2x^(-2) + 2ln|x| + C
