Question on: SS3 Mathematics - Integral Calculus (Integration)
Evaluate \(\int_{}^{}\frac{1}{x^{2}}dx\)
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A
\(\ln x^{2} + C\)
B
\(\frac{\ln x^{2}}{2x} + C \)
C
\( \frac{1}{2x} + C\)
D
\(\frac{2}{x^{2}} + C\ \)
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Correct Option: B
Let \(u = x^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore\frac{du}{dx} = 2x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore dx = \frac{du}{2x}\)
Substituting we have,
\(\int_{}^{}\frac{1}{x^{2}}\ dx = \ \int_{}^{}\frac{1}{u}.\frac{du}{2x}\ = \frac{1}{2x}\left( \ln u \right) + C\), substitute \(u\)
\[= \frac{1}{2x}\ln x^{2} + C\]
\(\therefore\int_{}^{}\frac{1}{x^{2}}dx = \frac{\ln x^{2}}{2x} + C\ \)
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