Question

The value of $\int \frac{e^x+9\cos x-2\sin x+7}{e^x+7\sin x+11\cos x+14}dx$ is
(where $c$ is the constant of integration)

• A. $\frac{1}{2}(x+\ln (e^x+7\sin x+11\cos x+14\left)\right)+c$
• B. $\frac{1}{2}(x-\ln (e^x+7\sin x+11\cos x+14\left)\right)+c$
• C. $x+\frac{1}{2}\ln (e^x+7\sin x+11\cos x+14))+c$
• D. $x-\frac{1}{2}\ln (e^x+7\sin x+11\cos x+14))+c$

Answer 1

The correct option is A $\frac{1}{2}(x+\ln (e^x+7\sin x+11\cos x+14\left)\right)+c$

$\int \frac{e^x+9\cos x-2\sin x+7}{e^x+7\sin x+11\cos x+14}dx$
$\int \frac{\frac{1}{2}(e^x+7\sin x+11\cos x+14)+\frac{1}{2}(e^x+7\cos x-11\sin x)}{(e^x+7\sin x+11\cos x+14)}dx$
$=\frac{1}{2}\int dx+\frac{1}{2}\int \frac{e^x+7\cos x-11\sin x}{(e^x+7\sin x+11\cos x+14)}dx$
$=\frac{x}{2}+\frac{1}{2}\ln (e^x+7\sin x+11\cos x+14)+c$