Question

Quizzz !!

[tex]\displaystyle\sf\int\limit_{-1}^1\:x.tan^{-1}x\:dx[/tex]

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Answer 1

Verified answer

PEMBAHASAN

Trigonometri

y = arctan x → y' = 1/(x² + 1)

tan⁻¹ x = arctan x

Integral parsial

∫u dv = uv - ∫v du

∫x arctan x dx

= ∫arctan x . x dx

= ∫arctan x d(x²)/2

= 1/2 x² arctan x - ∫(x²/2) d(arctan x)

= 1/2 x² arctan x - 1/2 ∫x² . 1/(x² + 1) dx

= 1/2 x² arctan x - 1/2 ∫(x²/(x² + 1)) dx

= 1/2 x² arctan x - 1/2 ∫(1 - 1/(x² + 1)) dx

= 1/2 x² arctan x - 1/2 (∫dx - ∫d(arctan x))

= 1/2 x² arctan x - 1/2 x + 1/2 arctan x

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tan 45° = 1

arctan 1 = 45° = π/4

tan (-45°) = -1

arctan (-1) = -π/4

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batas atas 1 , batas bawah (-1)

= (1/2 . 1² . π/4 - 1/2 . 1 + 1/2 . π/4) - (1/2 . (-1)² . (-π/4) - 1/2 . (-1) + 1/2 . (-π/4))

= (π/8 - 1/2 + π/8) - (-π/8 + 1/2 - π/8)

= 4π/8 - 1

= 1/2 π - 1