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If \( I=\int \frac{d x}{(2 x+5) \sqrt{4 x^{2}+20 x+16}}=\frac{1}{m} \sec ^{-1}\left(\frac{2 x+5}{p}\right)+c \) the |
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Answer» I = \(\int\cfrac{d\text x}{(2\text x+5)\sqrt{4\text x^2+20\text x+16}}\) = \(\int\cfrac{d\text x}{(2\text x+5)\sqrt{(2\text x+5)^2+9}}\) = \(\int\cfrac{d\text x}{(2\text x+5)\sqrt{(2\text x+5)^2+3^2}}\) = \(\cfrac12\times\cfrac13\) sec-1\(\left(\cfrac{2\text x+5}3\right)+c\) \(\left(\because\quad\int\cfrac{d\text x}{\text x\sqrt{\text x^2-a^2}}=\cfrac1asec^{-1}\cfrac{\text x}a+c\right)\) = \(\cfrac16\) sec-1\(\left(\cfrac{2\text x+5}3\right)+c\) Hence, m = 6, p = 3 |
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