Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals3 Marks
Question
Write a value of$\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$
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Answer
Let $\text{I}=\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$ Let $3+\text{a}^{\text{x}}=\text{t}$ $\text{a}^{\text{x}}\log\text{a dx}=\text{dt}$ $\text{a}^{\text{x}}\text{dx}=\frac{\text{dt}}{\log\text{a}}$ $\text{I}=\int\frac{\text{dt}}{\log\text{a}\cdot\text{t}}=\frac{1}{\log\text{a}}\log\text{t}+\text{C}$ $\text{I}=\frac{1}{\log\text{a}}\log(3+\text{a}^{\text{x}})+\text{C}$
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