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1-a-4-x-k-x-1-4a-3-Find-the-value-of-1-a-1-x-1-dx-




Question Number 129269 by bramlexs22 last updated on 14/Jan/21
 ∫_1 ^a  ((4(√x) +k)/( (√x) +1)) = 4a+3 . Find the value  of ∫_1 ^( a) (1/( (√x) +1)) dx .
$$\:\int_{\mathrm{1}} ^{{a}} \:\frac{\mathrm{4}\sqrt{{x}}\:+{k}}{\:\sqrt{{x}}\:+\mathrm{1}}\:=\:\mathrm{4}{a}+\mathrm{3}\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\int_{\mathrm{1}} ^{\:{a}} \frac{\mathrm{1}}{\:\sqrt{{x}}\:+\mathrm{1}}\:{dx}\:. \\ $$
Answered by liberty last updated on 14/Jan/21
 ∫_1 ^( a)  ((4(√x)+4+k−4)/( (√x)+1)) dx = 4a+3   ∫_1 ^( a) 4 dx +(k−4)∫_1 ^( a)  (dx/( (√x)+1))=4a+3   4a−4+(k−4)∫_1 ^( a)  (dx/( (√x)+1))=4a+3   ⇒ ∫_1 ^( a)  (dx/( (√x)+1))=(7/(k−4))
$$\:\int_{\mathrm{1}} ^{\:{a}} \:\frac{\mathrm{4}\sqrt{{x}}+\mathrm{4}+{k}−\mathrm{4}}{\:\sqrt{{x}}+\mathrm{1}}\:{dx}\:=\:\mathrm{4}{a}+\mathrm{3} \\ $$$$\:\int_{\mathrm{1}} ^{\:{a}} \mathrm{4}\:{dx}\:+\left({k}−\mathrm{4}\right)\int_{\mathrm{1}} ^{\:{a}} \:\frac{{dx}}{\:\sqrt{{x}}+\mathrm{1}}=\mathrm{4}{a}+\mathrm{3} \\ $$$$\:\mathrm{4}{a}−\mathrm{4}+\left({k}−\mathrm{4}\right)\int_{\mathrm{1}} ^{\:{a}} \:\frac{{dx}}{\:\sqrt{{x}}+\mathrm{1}}=\mathrm{4}{a}+\mathrm{3} \\ $$$$\:\Rightarrow\:\int_{\mathrm{1}} ^{\:\boldsymbol{{a}}} \:\frac{\boldsymbol{{dx}}}{\:\sqrt{\boldsymbol{{x}}}+\mathrm{1}}=\frac{\mathrm{7}}{\boldsymbol{{k}}−\mathrm{4}}\: \\ $$

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