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Integral hasil kali sinus-kosinus

Kalkulus Matematika · SMA hard

Hasil ∫4sin⁡5xcos⁡3x dx=…\displaystyle\int 4\sin 5x \cos 3x\,dx = \ldots

  1. A
    −2cos⁡8x−2cos⁡2x+C-2\cos 8x - 2\cos 2x + C
  2. B
    −14cos⁡8x−cos⁡2x+C-\tfrac14\cos 8x - \cos 2x + C
    ✓
  3. C
    14cos⁡8x+cos⁡2x+C\tfrac14\cos 8x + \cos 2x + C
  4. D
    −12cos⁡8x−2cos⁡2x+C-\tfrac12\cos 8x - 2\cos 2x + C
  5. E
    12cos⁡8x+2cos⁡2x+C\tfrac12\cos 8x + 2\cos 2x + C

Kunci jawaban

B

Penjelasan singkat

4sin⁡5xcos⁡3x=2(sin⁡8x+sin⁡2x)4\sin5x\cos3x=2(\sin8x+\sin2x); ∫=2(−cos⁡8x8)+2(−cos⁡2x2)=−14cos⁡8x−cos⁡2x+C\int=2\bigl(-\tfrac{\cos8x}{8}\bigr)+2\bigl(-\tfrac{\cos2x}{2}\bigr)=-\tfrac14\cos8x-\cos2x+C.

(Jawaban dihitung & diverifikasi dengan Python/SymPy.)

Gunakan 2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B=\sin(A+B)+\sin(A-B): 4sin⁡5xcos⁡3x=2(sin⁡8x+sin⁡2x)4\sin5x\cos3x=2\bigl(\sin8x+\sin2x\bigr).

∫(2sin⁡8x+2sin⁡2x) dx=−28cos⁡8x−22cos⁡2x+C=−14cos⁡8x−cos⁡2x+C\int(2\sin8x+2\sin2x)\,dx=-\tfrac{2}{8}\cos8x-\tfrac{2}{2}\cos2x+C=-\tfrac14\cos8x-\cos2x+C

→ B. (Turunan hasil ini diverifikasi kembali ke integran.)

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