The value of int_-pi/6^pi/6left(fracpi+4x^111-sin(|x|+pi/6)right)dx$\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\pi/6)}\right)dx$ is equal to

### Related Formula int_-a^a f(x) dx = int_0^a [f(x) + f(-x)] dx$$\int_{-a}^{a} f(x) dx = \int_{0}^{a} [f(x) + f(-x)] dx$$ ### Core Logic Let I = int_-pi/6^pi/6fracpi+4x^111-sin(|x|+pi/6)dx$I = \int_{-\pi/6}^{\pi/6}\frac{\pi+4x^{11}}{1-\sin(|x|+\pi/6)}dx$. The denominator 1 - sin(|x| + pi/6)$1 - \sin(|x| + \pi/6)$ is an even function. The numerator can be split into an even part (pi$\pi$) and an odd part (4x^11$4x^{11}$). int_-a^a frac4x^111-sin(|x|+pi/6) dx = 0 quad text(Since integrand is odd)$$\int_{-a}^{a} \frac{4x^{11}}{1-\sin(|x|+\pi/6)} dx = 0 \quad \text{(Since integrand is odd)}$$ ### Step 1: Simplify to Even Integral We are left with the even part: I = int_-pi/6^pi/6 fracpi1 - sin(|x| + pi/6) dx$$I = \int_{-\pi/6}^{\pi/6} \frac{\pi}{1 - \sin(|x| + \pi/6)} dx$$ Using even function property int_-a^a f(x) dx = 2 int_0^a f(x) dx$\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$: I = 2pi int_0^pi/6 frac11 - sin(x + pi/6) dx$$I = 2\pi \int_{0}^{\pi/6} \frac{1}{1 - \sin(x + \pi/6)} dx$$ ### Step 2: Substitution Let t = x + fracpi6 Rightarrow dt = dx$t = x + \frac{\pi}{6} \Rightarrow dt = dx$. Limits: when x = 0 Rightarrow t = pi/6$x = 0 \Rightarrow t = \pi/6$, when x = pi/6 Rightarrow t = pi/3$x = \pi/6 \Rightarrow t = \pi/3$. I = 2pi int_pi/6^pi/3 fracdt1 - sin t$$I = 2\pi \int_{\pi/6}^{\pi/3} \frac{dt}{1 - \sin t}$$ ### Step 3: Solve the Integral Rationalize the denominator: I = 2pi int_pi/6^pi/3 frac1 + sin t(1 - sin t)(1 + sin t) dt$$I = 2\pi \int_{\pi/6}^{\pi/3} \frac{1 + \sin t}{(1 - \sin t)(1 + \sin t)} dt$$ I = 2pi int_pi/6^pi/3 frac1 + sin tcos^2 t dt$$I = 2\pi \int_{\pi/6}^{\pi/3} \frac{1 + \sin t}{\cos^2 t} dt$$ I = 2pi int_pi/6^pi/3 (sec^2 t + sec t tan t) dt$$I = 2\pi \int_{\pi/6}^{\pi/3} (\sec^2 t + \sec t \tan t) dt$$ Integrate directly: I = 2pi left[ tan t + sec t right]_pi/6^pi/3$$I = 2\pi \left[ \tan t + \sec t \right]_{\pi/6}^{\pi/3}$$ Evaluate limits: Upper limit (pi/3$\pi/3$): tan(pi/3) + sec(pi/3) = sqrt3 + 2$\tan(\pi/3) + \sec(\pi/3) = \sqrt{3} + 2$ Lower limit (pi/6$\pi/6$): tan(pi/6) + sec(pi/6) = frac1sqrt3 + frac2sqrt3 = frac3sqrt3 = sqrt3$\tan(\pi/6) + \sec(\pi/6) = \frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$ I = 2pi [(sqrt3 + 2) - sqrt3] = 2pi (2) = 4pi$$I = 2\pi [(\sqrt{3} + 2) - \sqrt{3}] = 2\pi (2) = 4\pi$$ ### Pattern Recognition Symmetric limits [-a, a]$[-a, a]$ instantly demand testing for odd/even parity. Any mixed polynomial like c + k x^textodd$c + k x^{\text{odd}}$ over an even denominator guarantees the odd power term strictly vanishes, halving calculation time. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Definite Integration Class 11 Maths: Trigonometric Functions