Question

(i) By integrating by parts, show that ? x ln x dx = (x² ln x) / 2 - x² / 4 + C. If necessary, you may use the integral ? ln x dx = x ln x - x + C. Here C is an arbitrary constant. (ii) If u = e? / x, give a simplified expression for ln u. Hence use the substitution u = e? / x to find ? e²?(x - 1)(x - ln x) dx / x³

          (i) By integrating by parts, show that
? x ln x dx = (x² ln x) / 2 - x² / 4 + C.
If necessary, you may use the integral ? ln x dx = x ln x - x + C. Here C is an arbitrary constant.
(ii) If u = e? / x, give a simplified expression for ln u.
Hence use the substitution u = e? / x to find
? e²?(x - 1)(x - ln x) dx / x³

(i) By integrating by parts, show that
? x ln x dx = (x² ln x) / 2 - x² / 4 + C.
If necessary, you may use the integral ? ln x dx = x ln x - x + C. Here C is an arbitrary constant.
(ii) If u = e? / x, give a simplified expression for ln u.
Hence use the substitution u = e? / x to find
? e²?(x - 1)(x - ln x) dx / x³

Added by Lisa K.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Krishna G

Step 1

Let u = ln(x) and dv = x^2 dx. Then, du = (1/x) dx and v = (1/3)x^3. Using integration by parts, we have: ∫x^2 ln(x) dx = uv - ∫v du = (1/3)x^3 ln(x) - ∫(1/3)x^3 (1/x) dx = (1/3)x^3 ln(x) - (1/3)∫x^2 dx = (1/3)x^3 ln(x) - (1/3)(1/3)x^3 + C = (1/3)x^3 (ln(x) - Show more…

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By integrating by parts, show that ∫ x ln x dx = x² ln x / 2 - x² / 4 + C. If necessary, you may use the integral ∫ ln x dx = x ln x - x + C. Here C is an arbitrary constant. (ii) If u = eˣ / x, give a simplified expression for ln u. Hence use the substitution u = eˣ / x to find ∫ e²ˣ(x - 1)(x - ln x) dx / x³