Question
Evaluate the following integrals:
$\int\text{e}^{\text{x}}(\tan\text{x}-\log\cos\text{x})\text{dx}$
$\int\text{e}^{\text{x}}(\tan\text{x}-\log\cos\text{x})\text{dx}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\int\text{e}^{\text{x}}\log\cos\text{x dx}$
Integrating by parts
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\Big\{\text{e}^{\text{x}}\log\cos\text{x}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\log\cos\text{x}\Big)\text{dx}\Big\}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\big\{\text{e}^{\text{x}}\log\cos\text{x}+\int\text{e}^{\text{x}}\tan\text{x dx}\big\}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\text{e}^{\text{x}}\log\cos\text{x}-\int\text{e}^{\text{x}}\tan\text{x dx}+\text{C}$
$=-\text{e}^{\text{x}}\log\cos\text{x}+\text{C}$
$=\text{e}^{\text{x}}\log\sec\text{x}+\text{C}$ $\big[\because\log\sec\text{x}=-\log\cos\text{x}\big]$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| xi | 1 | 2 | 3 | 4 |
| pi | 0.4 | 0.1 | 0.2 | 0.3 |
$\sin(\log\text{x})$