If dx^3x√2 2x = ( x)^A + C( x)^B + k, where k is a constant of integration, then A+B+C equ…

Question

If $\displaystyle\int \dfrac{dx}{\cos^3x\sqrt{2\sin 2x}} = (	an x)^A + C(	an x)^B + k$, where $k$ is a constant of integration, then $A+B+C$ equals:

Accepted Answer

$\int \dfrac{dx}{\cos^3x\sqrt{4\sin x\cos x}} = \int \dfrac{dx}{2\cos^4x\sqrt{	an x}}$

Let $	an x = t^2 \Rightarrow \sec^2x\,dx = 2t\,dt$, and $\sec^2x = 1+t^4$

$= \int \dfrac{\sec^4x\,dx}{2\sqrt{	an x}} = \int \dfrac{\sec^2x(\sec^2x\,dx)}{2\sqrt{	an x}} = \int \dfrac{(1+t^4)\cdot 2t\,dt}{2t} = \int (1+t^4)\,dt = t+\dfrac{t^5}{5}+k$

$= \sqrt{	an x} + \dfrac15	an^{5/2}x + k$

$A = \dfrac12, B = \dfrac52, C = \dfrac15$

$A+B+C = \dfrac{16}{5}$

Solutions · 1