MIND FREK INTEGRAL(2)

\int \frac{cosx - sinx}{\sqrt{8-sin2x}}dx

ans-----> sin ^{-1}\left(\frac{sinx + cosx}{3} \right)+c

2 Answers

write sin2x as

( sinx + cosx ) ^2 = 1 + sin2x

sin2x = ( sinx + cosx ) ^2 - 1

\int \frac{cosx - sinx }{\sqrt{8 - [ ( sinx + cosx ) ^2 -1 )] }}

\int \frac{cosx - sinx }{\sqrt{9 - ( sinx + cosx ) ^2 }}

put sinx + cosx = t

cosx - sinx dx = dt

\int \frac{dt }{\sqrt{9 - ( t ) ^2 }} \rightarrow \sin^{-1}\left( \frac{t}{3}\right)

got it simple it was.i had not tried it.[4][4]

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω