Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals: $\int\tan^5\text{x}\text{ dx}$
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Answer
Let $\text{I}=\int\tan^5\text{x}\text{ dx}$ Then $\text{I}=\int\tan^2\text{x}\tan^3\text{x}\text{ dx}$ $=\int(\sec^2\text{x}-1)\tan^3\text{x}\text{ dx}$ $=\int\sec^2\text{x}\tan^3\text{x}\text{ dx}-\int\tan^3\text{x}\text{ dx}$ $=\int\sec^2\tan^3\text{x}\text{ dx}-\int(\sec^2\text{x}-1)\tan\text{x}\text{ dx}$ Substituting $\tan\text{x}=\text{t}$ and $\sec^2\text{x}\text{ dx}=\text{dt}$ in first two integral, we get $\text{I}=\int\text{t}^3\text{dt}-\int\text{tdt}+\int\tan\text{x}\text{ dx}$ $=\frac{\text{t}^4}{4}-\frac{\text{t}^2}{2}+\log|\sec\text{x}|+\text{C}$ $=\frac{\tan^4\text{x}}{4}-\frac{\tan^2\text{x}}{2}+\log|\sec\text{x}|+\text{C}$ $\therefore\ \text{I}=\frac{\tan^4\text{x}}{4}-\frac{\tan^2\text{x}}{2}+\log|\sec\text{x}|+\text{C}$
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