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Integral Calculus> plz help...

question mark

∫ xe^x cosx dx= f(x)+c, then find f(x).

PIYUSH AGRAWAL , 15 Years ago

Grade 12

1 Answers

Rathod Shankar AskiitiansExpert-IITB

Last Activity: 15 Years ago

1)

integral by parts take x and e^x cosx as two parts

$\int f g\, dx = f \int g\, dx - \int \left ( f' \int g\,dx \right )\, dx.\!$

take f=x and g=e^x cosx

here iam giving method for integration for e^x cosx ...i think u can solve now right. do practice so that u can get more concept

$\int e^{x} \cos (x) \, dx\!$

$u = \cos(x) \Rightarrow du = -\sin(x)\,dx$

$dv = e^x \, dx \Rightarrow v = \int e^x \, dx = e^x$

$\int e^{x} \cos (x) \, dx = e^{x} \cos (x) + \int e^{x} \sin (x) \, dx.\!$

$u = \sin(x) \Rightarrow du = \cos(x)\, dx$

$dv = e^x \, dx \Rightarrow v = \int e^x \, dx = e^x$

$\int e^x \sin (x) \, dx = e^x \sin (x) - \int e^x \cos (x) \,dx$

Putting these together,

$\int e^x \cos (x) \,dx = e^x \cos (x) + e^x \sin (x) - \int e^x \cos (x) \, dx.$

$2 \int e^{x} \cos (x) \, dx = e^{x} ( \sin (x) + \cos (x) ) + C\!$

$\int e^x \cos (x) \,dx = {e^x ( \sin (x) + \cos (x) ) \over 2} + C'\!$

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