Question

integral of e^x (1+sinx cosx)/cos^2x dx

Answer 1

Given, ∫ e^x (1+sinx cosx)/cos^2x dx

= ∫ e^x { 1 / cos^2 x + sinx.cosx / cos^2 x} dx

= ∫ e^x (tan x + sec^2 x) dx

= ∫ e^x tanx . dx + ∫ e^x ( sec^2x) dx

= tan x . e^x - ∫ sec^2 x.e^x dx + ∫ e^x .sec^2 xdx + C

= e^x tanx + C

Answer 2

YOUR QUESTION.

$\\ \ \qquad \huge{\cdot} \small\tt \int e^x \: \dfrac{(1+sin(x) \: cos(x))}{cos^2(x)} \: \: dx$

EXPLANATION.

$\\ \ \tt \: \huge{➯} \small \int e^x \: \bigg(\dfrac { 1} { cos^2( x)} + \dfrac{sin(x).cos(x)}{cos^2 (x)}\bigg) \: \: dx$

$\\ \ \tt \: \huge{➯} \small \int e^x \: \bigg(tan(x) + sec ^{2}(x) \bigg) \: \: dx$

$\\ \ \tt \: \huge{➯} \small \int e^x \: \bigg(tan(x) \cdot \: dx+ \int e^x \bigg(sec ^{2}(x) \bigg) \bigg) \: \: dx$

$\\ \ \tt \: \huge{➯} \: \small \: tan(x) \cdot \: {e}^{x} - \int sec ^{2}(x) \cdot \: e^x \: \: dx \: + \: \int \: {e}^{x} \cdot \: sec ^{2} \: (x) \: dx \: + \: c$

$\\ \ \: \bigstar \: \underline{\boxed{ \tt {\purple {\small \: {e}^{x} \: \: tan \: (x) \: + \: c}}}}$

$\underline{ \therefore \: \boldsymbol{Hence \: \rm \: the \:answer \: is \: \: { \pink {\boxed{\tt \: e {}^{x} \: \: tan(x) \: + \: c }}}}}.$

✰ Hope it helps u ✰