Question

Assertion :$\int e^x(\tan x+{\sec }^{2}x)dx=e^x\tan x+C$ Reason: $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+C$

• 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

$\int e^x(\tan x+{\sec }^{2}x)dx$
$=\int e^x\tan xdx+\int e^x{\sec }^{2}xdx$
$=\tan x\int e^{x}dx-\int \left(\frac{d}{dx}\right(\tan x)\int e^{x}dx)dx+\int e^x{\sec }^{2}xdx$
$=e^x\tan x-\int e^x{\sec }^{2}xdx+\int e^x{\sec }^{2}xdx$
$=e^x\tan x+C$
$\int e^x(\tan x+{\sec }^{2}x)dx=e^x\tan x+C$
Hence, assertion is correct.
Reason:$\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+C$
$\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=\int e^{x}f\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)-\int e^x{f}^{\prime }\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)+C$
Hence, reason is correct and also the explanation for assertion