Question
\(\int \frac{x^{2}+1}{(x^{2}+3x-5)^{2}}dx\)
A \(\int -\frac{3}{2}\frac{1}{(3x^{2}+3)^{2}} +c\)
B \(\int -\frac{1}{6}\frac{1}{(3x^{2}+3)^{2}} +c\)
C \(\int -\frac{3}{2}\frac{1}{x^{2}+3x-5)^{2}} +c\)
D \(\int -\frac{1}{6}\frac{1}{x^{2}+3x-5)^{2}} +c\)
▶️Answer/Explanation
Ans:D
Starting with the substitution \(u=x^{3}+3x-5,u=x^{3}+3x-5\Rightarrow \frac{\mathrm{d} u}{\mathrm{d} x}=3x^{2}+3=3(x^{2}+1)\Rightarrow dx=\frac{dy}{3(x^{2}+1)}\)
Substituting for \(x^{3}+3x-5\) and for
gives \(\int \frac{x^{2}+1}{x^{2}+3x-5}dx=\int \frac{1}{u^{3}}\frac{1}{3}du=\frac{1}{3}.(-\frac{1}{2u^{2}})+c=-\frac{1}{6}.\frac{1}{(x^{2}+)}\)
Question
\(\int \frac{xdx}{\sqrt{3x^{2}+5}}\)=
(A)\(\frac{1}{9}\left ( 3x^{2}+5 \right )^{\frac{3}{2}}+C\) (B)\(\frac{1}{4}\left ( 3x^{2}+5 \right )^{\frac{3}{2}}+C\) (C)\(\frac{1}{12}\left ( 3x^{2}+5 \right )^{\frac{1}{2}}+C\) (D)\(\frac{1}{3}\left ( 3x^{2}+5 \right )^{\frac{3}{2}}+C \) (E)\(\frac{3}{2}\left ( 3x^{2}+5 \right )^{\frac{1}{2}}+C\)
▶️Answer/Explanation
Ans:D
Question
\(\int_{0}^{\frac{\pi }{2}}\frac{cos\Theta }{\sqrt{1+sin\Theta }}d\Theta\)
(A)\(-2\left ( \sqrt{2}-1 \right ) \) (B)\(( -2\sqrt{2})\) (C)\(( -2\sqrt{2})\) (D)\(2(\sqrt{2}-1)\) (E)\(2(\sqrt{2}+1)\)
▶️Answer/Explanation
Ans:D
\(\int_{0}^{\frac{\pi}{2}}(1+sin \theta)^{-1/2}(cos \theta d \theta)=2(1+sin \theta)^{-1/2}|_{0}^{\frac{\pi}{2}}=2(\sqrt{2-1})\)
Question
\( \int_{2}^{3}\frac{x}{x^{2}+1}dx=\)
(A)\(\frac{1}{2}ln \frac{3}{2}\) (B)\(\frac{1}{2}ln2 \) (C) ln 2 (D) 2ln 2 (E)\(\frac{1}{2}ln5\)
▶️Answer/Explanation
Ans:B
