Question

find the anti derivative of
sec²(7-4x)​

Answer 1

check the attachment______♡

Answer 2

Answer

• ⁻¹/₄ tan(7-4x) + c

Given

• sec²(7-4x)

To Find

• Anti-derivative of given value

Solution

$\rm \int sec^2(7-4x)\ dx$

$\boxed{\begin{minipage}{5cm} \rm Let\ ,u=7-4x\\\\\implies \rm \dfrac{d}{dx}(u)=\dfrac{d}{dx}(7-4x)\\\\\implies \rm \dfrac{du}{dx}=\dfrac{d}{dx}(7)-4\dfrac{d}{dx}(x)\\\\\implies \rm \dfrac{du}{dx}=0-4\\\\\implies \rm du=-4dx\\\\\implies \rm dx=\dfrac{du}{-4} \end{minipage}}$

$\implies \rm \int sec^2(u)\ \dfrac{du}{-4}\\\\\implies \rm \dfrac{-1}{4} \int sec^2u\ du\\\\\implies \rm \dfrac{-1}{4}\ tanu+c\\\\\implies \bf \dfrac{-1}{4}\ tan( 7-4x)+c\ \;\; \bigstar$

Formula Used

$\bullet \;\; \bf \dfrac{d}{dx}(tan\ x)=sec^2x\\\\\bullet \;\; \bf \int sec^2x=tan\ x+c$