Mathematics, 03.02.2020 00:46 cxttiemsp021

Find integrate (1/(xsqrt(1+(lnx),e, x)

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Note that x² + 2x + 3 = x² + x + 3 + x. So your integrand can be written as

(x² + x + 3 + x)/(x² + x + 3) = 1 + x/(x² + x + 3).

Next, complete the square.

x² + x + 3 = x² + x + 1/4 + 11/4 = (x + 1/2)² + (√(11)/2)²

Also, for the x in the numerator

x = x + 1/2 - 1/2.

So

(x² + 2x + 3)/(x² + x + 3) = 1 + (x + 1/2)/[(x + 1/2)² + (√(11)/2)²] - 1/2/[(x + 1/2)² + (√(11)/2)²].

Integrate term by term to get

∫ (x² + 2x + 3)/(x² + x + 3) dx = x + (1/2) ln(x² + x + 3) - (1/√(11)) arctan(2(x + 1/2)/√(11)) + C

b) Use the fact that ln(x) = 2 ln√(x). Then put u = √(x), du = 1/[2√(x)] dx.

∫ ln(x)/√(x) dx = 4 ∫ ln u du = 4 u ln(u) - u + C = 4√(x) ln√(x) - √(x) + C

= 2 √(x) ln(x) - √(x) + C.

c) There are different approaches to this. One is to multiply and divide by e^x, then use u = e^x.

∫ 1/(e^(-x) + e^x) dx = ∫ e^x/(1 + e^(2x)) dx = ∫ du/(1 + u²) = arctan(u) + C

= arctan(e^x) + C.

Answer from: Quest

step-by-step explanation:

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Answer from: Quest

answer: 2 answers

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