Integral with Trigonometric function.

JEE Advanced 2015 Online › Calculus questions › Integral with Trigonometric function.

\hspace{-16}(1):\; \mathbf{\int x\sqrt{1+\sin x}dx}\\\\\\ (2):\; \mathbf{\int_{0}^{\pi}\frac{2+2(x+1)\sin x-(x^2+1)\cos^2 x}{\sin x-x\cos x+1}dx}\\\\\\ (3):\;\mathbf{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos^2 x}{(e^x+1)(x\sin x+\cos x)^2}dx}\\\\\\

#Mathematics #Calculus

UP 0 DOWN 0 0 2
Post on FacebookPost anonymously?

2 Answers

Shubhodip ·2011-07-31 10:31:21

Let it be I₃

doing f(a+b-x) we get 2I₃=(same lims as given) âˆ«cos²x(xsinx+ cosx)²

but âˆ«cos²x(xsinx+ cosx)² = sinxxsinx + cos x+ c (well known tric P(x)/q(x) rule)

UP 0 DOWN 0
Post on FacebookPost anonymously?

narayan ·2011-08-01 00:20:56

I=\int x\sqrt{\left ( sin\frac{x}{2} +cos\frac{x}{2}\right )^2} dx

\int x{\left ( sin\frac{x}{2} +cos\frac{x}{2}\right )} dx

\int x\left sin\frac{x}{2}dx +\int xcos\frac{x}{2}\right dx

\texttt{using integration by parts we get }

\int x\left sin\frac{x}{2}dx =-2xcos\left (\frac{x}{2}\right )+ 4sin\left ( \frac{x}{2} \right )

\int x\left cos\frac{x}{2}dx =2xsin\left (\frac{x}{2}\right )+ 4cos\left ( \frac{x}{2} \right )

I =\int x\left cos\frac{x}{2}dx+\int x\left sin\frac{x}{2}dx=2xsin\left (\frac{x}{2}\right )+ 4cos\left ( \frac{x}{2} \right )-2xcos\left ( \frac{x}{2} \right )+4sin\left ( \frac{x}{2} \right )

\boxed{\boxed{I =2(x+2)sin\left (\frac{x}{2}\right ) -2(x-2)cos\left ( \frac{x}{2} \right )}}

UP 0 DOWN 0
Post on FacebookPost anonymously?

Your Answer

Basic Integral Calculus Operators Symbols Matrix Cases

√ √ || {} [] () → ~ ^ 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

Close [X]

Integral with Trigonometric function.

2 Answers

Your Answer

Add Image

Similar Questions