Answer to Question #250544 in Calculus for JaytheCreator

Question #250544

Find (i) ∫ 𝑑π‘₯ π‘₯(ln π‘₯)2

(ii) ∫ π‘₯√π‘₯ + 1 𝑑π‘₯

(iii) ∫ π‘₯(π‘₯ 2 + 3) 4𝑑π‘₯ 1 0

(iv) cos2 π‘₯ sin3 π‘₯ 𝑑π‘₯. Try 𝑑 = sin π‘₯Β 


1
Expert's answer
2021-10-13T17:48:17-0400

Solution:

(i) "I=\\int \\dfrac{dx}{x (\\ln x)^2}" ...(i)

Put "\\ln x=t"

"\\Rightarrow\\dfrac 1xdx=dt"

So, (i) becomes,

"I=\\int \\dfrac{dt}{ (t)^2}\n\\\\=\\int t^{-2}dt\n\\\\=-t^{-1}+C\n\\\\=-\\dfrac 1t+C\n\\\\=-\\dfrac 1{\\ln x}+C"

(ii) "I=\\int \\dfrac{x}{\\sqrt{x+1}}dx" ...(ii)

Put "x+1=t"

"\\Rightarrow dx=dt"

So, (ii) becomes,

"I=\\int \\dfrac{t-1}{\\sqrt t}dt\n\\\\=\\int (\\dfrac{t}{\\sqrt t}-\\dfrac{1}{\\sqrt t})dt\n\\\\=\\int( t^{1\/2}-t^{-1\/2})dt\n\\\\=\\dfrac{t^{3\/2}}{3\/2}-\\dfrac{t^{1\/2}}{1\/2}+C\n\\\\=\\dfrac 23(x+1)^{3\/2}-2(x+1)^{1\/2}+C"

(iii) "I=\\int \\dfrac {x}{(x^2+3)^4}dx" ...(iii)

Put "x^2+3=t"

"\\Rightarrow 2xdx=dt\n\\\\\\Rightarrow xdx=\\dfrac{dt}2"

So, (iii) becomes,

"I=\\dfrac 12\\int \\dfrac {dt}{(t)^4}\n\\\\=\\dfrac 12\\int t^{-4}{dt}\n\\\\=\\dfrac 12(\\dfrac{t^{-3}}{-3})+C\n\\\\=-\\dfrac 1{6(x^2+3)^3}+C"

(iv) "I=\\int \\cos2x\\sin 3x\\ dx"

"=\\dfrac12\\int (2 \\cos2x\\sin 3x)dx\n\\\\=\\dfrac12\\int (\\sin(2x+3x)-\\sin(2x-3x))dx\n\\\\=\\dfrac12\\int (\\sin(5x)-\\sin(-x))dx\n\\\\=\\dfrac12\\int (\\sin(5x)+\\sin(x))dx\n\\\\=\\dfrac12 (-\\dfrac{\\cos(5x)}5-\\cos x)+C"


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