To determine

To evaluate:

The indefinite integral ∫sec2⁡2θ dθ

Answer

12tan⁡2θ+C

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du.

ii) Indefinite integral

∫sec2⁡x dx=tan⁡x+C

2) Given:

∫sec2⁡2θ dθ

3) Calculation:

Here, use the substitution method because by substituting u=2θ the given integral become simpler integral.

Substitute u=2θ.

Differentiate u=2θ with respect to θ.

du=2dθ

Divide by 2 on both sides.

12du=dθ

By using concept i),

substitute u=2θ ,  dθ=12du to get.

∫sec2⁡2θ dθ=∫sec2⁡u12du=12∫sec2⁡udu

By using concept ii),

=12tan⁡u+C

Resubstitute u=2θ.

=12tan⁡2θ+C

Conclusion:

Therefore,

∫sec2⁡2θ dθ=12tan⁡2θ+C