Evaluate the integral

Ideas required to solve the problems:

* Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integration. If derivative of a function is present in an integration or if chances of its presence after few modification is possible then we apply integration by substitution method.

* Knowledge of integration of fundamental functions like sin, cos, polynomial, log etc and formula for some special functions.

Let, $I=\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x$

To solve such integrals involving trigonometric terms in numerator and denominators. We use the basic substitution method and to apply this simply we follow the undermentioned procedure-

If I has the form $\int \frac{a \sin x+b \cos x+c}{d \sin x+e \cos x+f} d x$

Then substitute numerator as -

$a \sin x+b \cos x+c=A \frac{d}{d x}(d \sin x+e \cos x+f)+B(d \sin x+e \cos x+c)+C$

Where A, B and C are constants

We have, $I=\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x$

As I matches with the form described above, So we will take the steps as described.

$\therefore 4 \sin x+5 \cos x=A \frac{d}{d x}(5 \sin x+4 \cos x)+B(4 \cos x+5 \sin x)+C$

$\Rightarrow 4 \sin x+5 \cos x=A(5 \cos x-4 \sin x)+B(4 \cos x+5 \sin x)+C \quad\left\{\because \frac{d}{d x} \cos x=-\sin x\right\}$

$\Rightarrow 4 \sin x+5 \cos x=\sin x(5 B-4 A)+\cos x(5 A+4 B)+C$

Comparing both sides we have:

$C=0$

$5 B-4 A=4$

$4 B+5 A=5$

On solving for $A, B$ and $C$ we have:

$A=9 / 41, B=40 / 41$ and $C=0$

Thus I can be expressed as:

$I=\int \frac{\frac{9}{41}(5 \cos x-4 \sin x)+\frac{40}{41}(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$

$I=\int \frac{\frac{9}{41}(5 \cos x-4 \sin x)}{4 \cos x+5 \sin x} d x+\int \frac{\frac{40}{41}(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$

$\therefore$ Let $\mathrm{I}_{1}=\frac{9}{41} \int \frac{(5 \cos \mathrm{x}-4 \sin \mathrm{x})}{4 \cos \mathrm{x}+5 \sin \mathrm{x}}$ and $\mathrm{I}_{2}=\frac{40}{41} \int \frac{(4 \cos \mathrm{x}+5 \sin \mathrm{x})}{4 \cos \mathrm{x}+5 \sin \mathrm{x}} \mathrm{dx}$

$\Rightarrow \mathrm{I}=\mathrm{I}_{1}+\mathrm{I}_{2} \ldots$ equation 1

$I_{1}=\frac{9}{41} \int \frac{(5 \cos x-4 \sin x)}{4 \cos x+5 \sin x}$

Let, $4 \cos x+5 \sin x=u$

$\Rightarrow(-4 \sin x+5 \cos x) d x=d u$

So, $I_{1}$ reduces to:

$I_{1}=\frac{9}{41} \int \frac{\mathrm{du}}{\mathrm{u}}=\frac{9}{41} \log |\mathrm{ul}|+C_{1}$

$\therefore \mathrm{I}_{1}=\frac{9}{41} \log |4 \cos \mathrm{x}+5 \sin \mathrm{x}|+\mathrm{C}_{1} \ldots \ldots$ equation 2

As, $I_{2}=\frac{40}{41} \int \frac{(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$

$\Rightarrow I_{2}=\frac{40}{41} \int d x=\frac{40 x}{41}+C_{2} \ldots . .$ equation 3

From equation 1,2 and 3 we have:

$I=\frac{9}{41} \log |4 \cos x+5 \sin x|+C_{1}+\frac{40 x}{41}+C_{2}$

$\therefore I=\frac{9}{41} \log |4 \cos x+5 \sin x|+\frac{40 x}{41}+C$