A) \[\sqrt{{{x}^{2}}+10x+24}+{{\sec }^{-1}}(x+5)+C\]
B) \[\sqrt{{{x}^{2}}+10x+24}-cose{{c}^{-1}}(x+5)+C\]
C) \[{{\sec }^{-1}}(x+5)-\sqrt{{{x}^{2}}+10x+24}+C\]
D) \[\cos e{{c}^{-1}}(x+5)+\sqrt{{{x}^{2}}+10x+24}+C\]
Correct Answer: D
Solution :
[d] \[I=\int{\frac{\sqrt{{{x}^{2}}+10x+24}}{x+5}}\,\,dx=\int{\frac{\sqrt{{{(x+5)}^{2}}-1}}{x+5}}\,dx\] Put \[x+5=\sec \theta \Rightarrow dx=\sec \theta \tan \theta \,d\theta .\] Then \[I=\int{\frac{\tan \theta }{\sec \theta }}.\sec \theta \tan \theta d\theta \] \[=\int{({{\sec }^{2}}\theta -1)\,d\theta }\] \[=\tan \theta -\theta +C\] \[=\sqrt{{{x}^{2}}+10x+24}-{{\sec }^{-1}}(x+5)+C\] \[=\text{cose}{{\text{c}}^{-1}}(x+5)+\sqrt{{{x}^{2}}+10x+24}+C\]
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