Evaluation of $\int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx$

Question

Evaluation of $\displaystyle\int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx$

$\bf{My\; Try::}$Let $$\displaystyle I = \int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx = \int\left(\frac{a+b\cos x}{b+a\cos x}\right)\cdot \frac{2\sin x}{(b+a\cos x)^2}dx$$

Now Let $$\displaystyle \left(\frac{a+b\cos x}{b+a\cos x}\right) = t\;,$$ Then $$\displaystyle \frac{\left[a^2-b^2\right]\sin x}{(b+a\cos x)^2}dx = dt\Rightarrow \frac{\sin x}{(b+a\cos x)^2}dx = \frac{1}{(a^2-b^2)}dt$$

So Integral $$\displaystyle I = \frac{2}{(a^2-b^2)}\int tdt = \frac{1}{(a^2-b^2)}\cdot \left(\frac{a+b\cos x}{b+a\cos x}\right)^2+\mathcal{C}$$

My Question is can we solve it any other Substution, If yes then plz explain here

Thanks

Answer 1

can as follows:

\begin{align} I &= \int {\frac{{2a\sin x + b\sin 2x}}{{\left( {b + a\cos x} \right)^3 }}dx} = 2a\int {\frac{{\sin x}}{{\left( {b + a\cos x} \right)^3 }}dx} + 2b\int {\frac{{\sin x\cos x}}{{\left( {b + a\cos x} \right)^3 }}dx} \\ \end{align} Let $u = \cos x \Rightarrow du = - \sin xdx $ \begin{align} I &= 2a\int {\frac{{\sin x}}{{\left( {b + au} \right)^3 }}\frac{{du}}{{ - \sin x}}} + 2b\int {\frac{{u\sin x}}{{\left( {b + au} \right)^3 }}\frac{{du}}{{ - \sin x}}} \\ &= 2a\int {\frac{1}{{\left( {b + au} \right)^3 }}du} - 2b\int {\frac{u}{{\left( {b + au} \right)^3 }}du} \\ &= 2a\int {\left( {b + au} \right)^{ - 3} du} - 2b\int {u\left( {b + au} \right)^{ - 3} du} \end{align} The first integral can be evaluated directly and the second one using integration by parts we get: \begin{align} I &= - \left( {b + au} \right)^{ - 2} + \frac{b}{a}\frac{u}{{\left( {b + au} \right)^2 }} + \frac{b}{{a^2 \left( {b + au} \right)}} \\ &= - \left( {b + a \cos x } \right)^{ - 2} + \frac{b}{a}\frac{\cos x}{{\left( {b + a\cos x} \right)^2 }} + \frac{b}{{a^2 \left( {b + a\cos x} \right)}}. \end{align}

Answer 2

Multiply with $\cos x$. Then $a2\cos x \sin x+b\cos x \sin 2x=\sin 2x(a+b\cos x)$.

Answer 3

$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$

Using Same process..

Let $$\displaystyle I = \int\frac{2a\sin x+b\sin 2x}{(b+a\cos x)^3}dx\;,$$ Now Divide both $\bf{N_{r}}$ and $\bf{D_{r}}$ by $\sin^3 x\;,$ We get

$$\displaystyle I = \int\frac{2a\csc^2 x+2b\cot x\cdot \csc x}{(b\csc x+a\cot x)^3}dx$$

Now Let $(b\csc x+a\cot x) =t\;,$ Then $\left(b\csc x\cdot \cot^2 x+a\csc^2 x\right)dx = -dt$

So Integral $$\displaystyle I = -2\int\frac{1}{t^3}dt = \frac{1}{t^2}+\mathcal{C} = \frac{1}{(b\csc x+a\cot x)^2}+\mathcal{C}$$