∫secxatanx+bdx is equal to
1a2+b2logtanπ4−x2+12tan−1ab+K
1a2+b2logtanπ4−x2−12tan−1ab+C
1a2+b2logtanπ4+x2−12tan−1ab+K
1a2+b2logtanπ4+x2+12tan−1ab+C
Let J=∫secxatanx+bdx
=∫1asinx+bcosxdx=∫1a2+b2aa2+b2sinx+ba2+b2cosxdx Choose sinα=aa2+b2,cosα=ba2+b2,tanα=ab⇒α=tan−1ab
J=1a2+b2∫1sinαsinx+cosαcosxdxJ=1a2+b2∫1cos(x−α)dx=1a2+b2∫sec(x−α)dxJ=1a2+b2logtanπ4+x−α2+CJ=1a2+b2logtanπ4+x2−12tan−1ab+K