Answer to Question #247643 in Calculus for JaytheCreator

Question #247643

Question 1 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 x


1
Expert's answer
2021-10-11T06:01:32-0400

"\\displaystyle\n2\\int x\\ln x \\\\\n\\text{Using integration by parts, we evaluate the above integral}\\\\\n\\text{Let $u = \\ln x, dv = x$ }\\implies du = \\frac{1}{x}, v = \\frac{x^2}{2}\\\\\n\\int udv = uv -\\int vdu\\\\\n\\implies 2\\int x\\ln x = 2(\\frac{x^2}{2}\\ln x - \\frac{x^2}{4}+c)\\\\\nx^2\\ln x - \\frac{x^2}{2}+c\\\\\n\\int (x\\sqrt{x}+1)dx = \\int (x^{\\frac{3}{2}}+1)dx\\\\\n= \\frac{2x^{\\frac{5}{2}}}{5}+x+c\\\\\n\\int x(x^2+3)4dx= \\int (4x^3 +12x)dx\\\\\nx^4+6x^2+c\\\\\n\\int \\cos 2x \\sin 3x dx\\\\\n\\sin A \\cos B = \\frac{1}{2}(\\sin 5x + \\sin x)dx\\\\\n\\therefore \\int \\cos 2x \\sin 3x dx = \\frac{1}{2}\\int \\sin 5xdx +\\frac{1}{2} \\int \\sin xdx\\\\\n=-\\frac{1}{10}\\cos5x-\\frac{1}{2}\\cos x+c"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS