5.3 fundamental theorem of calculus\n9. find the following definite integrals using the fundamental theorem…

5.3 fundamental theorem of calculus\n9. find the following definite integrals using the fundamental theorem of calculus.\na. $\\int_{1}^{e}\\left(\\frac{5}{x}+x\\right) d x$\nd. $\\int_{-1}^{1} \\frac{4 d x}{1+x^{2}}$\nb. $\\int_{0}^{\\pi}(2 \\cos (\\theta)-3 \\sin (\\theta)) d \\theta$\ne. $\\int_{1}^{64}\\left(z^{-2 / 3}+4 \\sqrt3{z}\\right) d z$\nc. $\\int_{4}^{81}\\left(\\frac{5}{\\sqrt{x}}+7\\right) d x$\n

5.3 fundamental theorem of calculus\n9. find the following definite integrals using the fundamental theorem of calculus.\na. $\\int_{1}^{e}\\left(\\frac{5}{x}+x\\right) d x$\nd. $\\int_{-1}^{1} \\frac{4 d x}{1+x^{2}}$\nb. $\\int_{0}^{\\pi}(2 \\cos (\\theta)-3 \\sin (\\theta)) d \\theta$\ne. $\\int_{1}^{64}\\left(z^{-2 / 3}+4 \\sqrt3{z}\\right) d z$\nc. $\\int_{4}^{81}\\left(\\frac{5}{\\sqrt{x}}+7\\right) d x$\n

Answer

Explanation:

Step1: Find antiderivative

For (\int\left(\frac{5}{x}+x\right)dx), the antiderivative of (\frac{5}{x}) is (5\ln|x|) (since (\int\frac{1}{x}dx=\ln|x|+C)) and the antiderivative of (x) is (\frac{x^{2}}{2}) (using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C,n\neq - 1)). So the antiderivative (F(x)=5\ln x+\frac{x^{2}}{2}) (since (x>0) in the interval ([1,e])).

Step2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that (\int_{a}^{b}f(x)dx=F(b)-F(a)). Here (a = 1), (b=e). [ \begin{align*} F(e)-F(1)&=\left(5\ln e+\frac{e^{2}}{2}\right)-\left(5\ln1+\frac{1^{2}}{2}\right)\ &=(5\times1+\frac{e^{2}}{2})-(5\times0+\frac{1}{2})\ &=5+\frac{e^{2}}{2}-\frac{1}{2}\ &=\frac{10 + e^{2}-1}{2}\ &=\frac{e^{2}+9}{2} \end{align*} ]

Answer:

(\frac{e^{2}+9}{2})