4.10 antiderivatives\n8. find the following most general antiderivatives (indefinite integrals).\na…

4.10 antiderivatives\n8. find the following most general antiderivatives (indefinite integrals).\na. $\\int\\left(2 e^{x}-9 x^{2}+x^{3 / 5}\\right) d x$\ne. $\\int\\left(\\frac{3 x^{4}+1}{x^{2}}-\\frac{1}{\\sqrt{x}}\\right) d x$\nb. $\\int\\left(3 \\cos (x)+2 \\sec ^{2}(x)\\right) d x$\nf. $\\int\\left(x^{2}+3\\right)(2 x+1) d x$\nc. $\\int\\left(\\sec (\\theta) \\tan (\\theta)+\\frac{3}{\\theta}\\right) d \\theta$\ng. $\\int\\left(9 e^{-3 x}-4 \\sin (2 x)\\right) d x$\n

4.10 antiderivatives\n8. find the following most general antiderivatives (indefinite integrals).\na. $\\int\\left(2 e^{x}-9 x^{2}+x^{3 / 5}\\right) d x$\ne. $\\int\\left(\\frac{3 x^{4}+1}{x^{2}}-\\frac{1}{\\sqrt{x}}\\right) d x$\nb. $\\int\\left(3 \\cos (x)+2 \\sec ^{2}(x)\\right) d x$\nf. $\\int\\left(x^{2}+3\\right)(2 x+1) d x$\nc. $\\int\\left(\\sec (\\theta) \\tan (\\theta)+\\frac{3}{\\theta}\\right) d \\theta$\ng. $\\int\\left(9 e^{-3 x}-4 \\sin (2 x)\\right) d x$\n

Answer

Explanation:

Step1: Integrate each term separately

Use the rules (\int e^{x}dx = e^{x}+C), (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)) For (\int(2e^{x}-9x^{2}+x^{3/5})dx), we have: (\int2e^{x}dx-\int9x^{2}dx+\int x^{3/5}dx)

Step2: Apply the integral formulas

(\int2e^{x}dx = 2e^{x}), (\int9x^{2}dx=9\times\frac{x^{3}}{3}=3x^{3}), (\int x^{3/5}dx=\frac{x^{\frac{3}{5}+1}}{\frac{3}{5}+1}=\frac{x^{\frac{8}{5}}}{\frac{8}{5}}=\frac{5}{8}x^{\frac{8}{5}})

Answer:

(2e^{x}-3x^{3}+\frac{5}{8}x^{\frac{8}{5}}+C)