What is the integral of tan4x?
Find the integral of tan4x
∫tan4xdx=∫tan4x-1+1dx⇒∫tan4xdx=∫tan2x2-122+1dx⇒∫tan4xdx=∫tan2x+12tan2x-12+1dx⇒∫tan4xdx=∫sec2xtan2x-1+1dx∵sec2x-tan2x=1⇒∫tan4xdx=∫sec2xtan2x-1dx+∫1dx⇒∫tan4xdx=∫sec2xtan2x-1dx+x+CwhereCisaconstant...(1)
Taketanx=t⇒sec2xdx=dt.
Put the above values in the equation(1):
∫tan4xdx=∫t2-1dt+x+C⇒∫tan4xdx=t33-t+x+C⇒∫tan4xdx=tan3x3-tanx+x+CPutt=tanx
Hence, the required integral is ∫tan where C is a constant of integration .
What is the geometrical interpretation of indefinite integral?
What is integral of (ax+b)/(cx+d)