Question

Assertion :$\int \frac{10x^9+{10}^x{\log }_{e}10}{{10}^x+x^{10}}dx=\log |{10}^x+x^{10}|+C$ Reason: $\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=\log \left|f\left(x\right)\right|+C$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\int \frac{10x^9+{10}^x{\log }_{e}10}{{10}^x+x^{10}}dx=\log |{10}^x+x^{10}|+C$
Consider,$\int \frac{10x^9+{10}^x{\log }_{e}10}{{10}^x+x^{10}}dx$
Let ${10}^x+x^{10}=t$
$\therefore$ $\frac{d}{dx}({10}^x+x^{10})=\frac{dt}{dx}$
$\therefore$ $({10}^x{\log }_{e}10+10x^9)dx=dt$
$\therefore$ Given integral $=\int \frac{dt}{t}=\log \left|t\right|+C$
$=\log |{10}^x+x^{10}|+C$
Hence assertion is true.
$\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=\log \left|f\left(x\right)\right|+C$
Reason is true and is the correct explanation for assertion