Question

∫[(sin2x)/(sin⁴x + cos⁴x)]dx = ______ + c

(a) tan^–1[√(tanx)] 

(b) tan^–1[(1/2)(tanx)]

(c) tan^–1(tan²x)

(d) tan^–1(2tanx)

Answer 1

The correct option (c) tan^–1(tan²x)   

Explanation:

∫[(sin2x dx)/(sin⁴x + cos⁴x)] = ∫[(2sinx cosx dx)/(sin⁴x + cos⁴x)]

= ∫[(2 tanx ∙ sec²x)/(1 + tan⁴x)]dx

Let   tan²x = t

∴   2tanx ∙ sec²x ∙ dx = dt

∴   I = ∫[dt/(1 + t²)] = tan^–1(t) + c

∴    I = tan^–1(tan²x) + c