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Question

Mathematics Question on integral

∫ex(1+x)cos2(exx) dx equals∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx \ equals

A

−cot(exx)+C- cot (e^xx) +C

B

tan(xex)+Ctan (xe^x) +C

C

tan(ex)+Ctan (e^x) +C

D

cot(ex)+Ccot (e^x) +C

Answer

tan(xex)+Ctan (xe^x) +C

Explanation

Solution

∫ex(1+x)cos2(exx) dx∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx

Let exx = t

⇒(ex. x+ ex.1)dx = dt

ex (x+1)dx = dt

∴ ∫ex(1+x)cos2(exx) dx∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx = ∫dtcos2t∫\frac {dt}{cos^2 t}

= ∫sec2t dt∫sec^2 t\ dt

= tan t+Ctan\ t+C

= tan (ex.x)+Ctan\ (e^x. x)+C

= tan (xex)+Ctan\ (xe^x)+C

Hence, the correct Answer is (B): tan (xex)+Ctan\ (xe^x)+C