definite integrals.. plz help

plz post the solutions for these questions

1) find the value of \int_{0}^{100}{[tan^{-1}x]} where [] equals gretest integer function

2)find the value of \int_{-2}^{2}{min {x-[x],-x-[-x]}dx } where [] denotes greatest integer function

3)\int_{1}^{2}{(x^{[x^{2}]}+[x^{2}]^{2})dx}

2 Answers

Ans 1 \int_{0}^{100}{[tan^{-1}x]} = \int_{0}^{tan1}{0dx} + \int_{tan1}^{100}{dx} = 100 - tan1

Ans 3\int_{1}^{\sqrt{2}}({x+1})dx + \int_{\sqrt{2}}^{\sqrt{3}}({x^{2}+4})dx + \int_{\sqrt{3}}^{2}({x^{3} +9})dx

Ans 2 ..use the property that [-x] = -(1+[x]) and x -[x] = {x}

so u will get the function as

f(x) = min( {x} , 1 - {x} )

now u wll get four triangles on the x axis with base 1 unit and height 0.5 unit ..so total area = 1 unit²

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

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

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

° ∂ ∞ 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