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Hasil dari ∫ (2x-1)√(x²-x+5) dx

Kalkulus Matematika · SMA medium

Hasil dari ∫(2x−1)x2−x+5 dx=....\displaystyle\int (2x-1)\sqrt{x^2-x+5}\,dx=....

  1. A
    12(x2−x+5)x2−x+5+C\tfrac{1}{2}(x^2-x+5)\sqrt{x^2-x+5}+C
  2. B
    23(x2−x+5)x2−x+5+C\tfrac{2}{3}(x^2-x+5)\sqrt{x^2-x+5}+C
    ✓
  3. C
    (x2−x+5)x2−x+5+C(x^2-x+5)\sqrt{x^2-x+5}+C
  4. D
    32(x2−x+5)x2−x+5+C\tfrac{3}{2}(x^2-x+5)\sqrt{x^2-x+5}+C
  5. E
    2(x2−x+5)x2−x+5+C2(x^2-x+5)\sqrt{x^2-x+5}+C

Kunci jawaban

B

Penjelasan singkat

Misal u=x2−x+5u=x^2-x+5, du=(2x−1) dxdu=(2x-1)\,dx. ∫u du=23u3/2=23(x2−x+5)x2−x+5+C\int\sqrt{u}\,du=\tfrac23 u^{3/2}=\tfrac23(x^2-x+5)\sqrt{x^2-x+5}+C.

Substitusi u=x2−x+5⇒du=(2x−1) dxu=x^2-x+5\Rightarrow du=(2x-1)\,dx.

∫(2x−1)x2−x+5 dx=∫u du=23u3/2+C=23(x2−x+5)x2−x+5+C\int(2x-1)\sqrt{x^2-x+5}\,dx=\int\sqrt{u}\,du=\tfrac{2}{3}u^{3/2}+C=\tfrac{2}{3}(x^2-x+5)\sqrt{x^2-x+5}+C.

Jawabannya B.

⚡ Latihan Matematika SMA