L13: Inverse Trig Antiderivatives Overview

Overview

Lecture covers integrals that yield or use inverse trigonometric functions, primarily via substitution. Includes core antiderivative rules and worked examples.

Elementary Antiderivatives (Inverse Trig)

• Core idea: Use known derivatives of inverse trig to identify antiderivatives.
• No separate need for arccos rule since it differs from arcsin by a sign and constant.

Antiderivative–Derivative Pairs

• arcsin(x) and −arccos(x) differ by a constant; same integrand yields either up to C.
• Similar relationship holds for arccsc and arcsec.

Example A: ∫ 1/(9 + x^2) dx

• Factor 9: ∫ 1/[9(1 + (x^2/9))] dx = (1/9) ∫ 1/(1 + (x/3)^2) dx.
• Substitute u = x/3, du = (1/3) dx; adjust with factor 3.
• Result: (1/3) arctan(x/3) + C.

Example B: ∫ 1/sqrt(25 − 4x^2) dx

• Factor 25 inside root: sqrt(25(1 − (4x^2/25))) = 5 sqrt(1 − (2x/5)^2).
• Pull 1/5 outside: (1/5) ∫ 1/sqrt(1 − (2x/5)^2) dx.
• Substitute u = (2/5)x, du = (2/5) dx; adjust with 5/2.
• Result: (1/2) arcsin(2x/5) + C.

Example C: ∫ e^x / [e^x sqrt(e^{2x} − 1)] dx

• Note e^{2x} = (e^x)^2; let u = e^x, du = e^x dx.
• Integrand becomes ∫ 1/[u sqrt(u^2 − 1)] du.
• Since u = e^x > 0, |u| = u; apply arcsec rule.
• Result: arcsec(e^x) + C.
• Remark: Cancelling e^x yields ∫ 1/sqrt(e^{2x} − 1) dx = arcsec(e^x) + C.

Example D: ∫ 1/[x + x(ln x)^2] dx

• Factor x: ∫ [1/x] · [1/(1 + (ln x)^2)] dx.
• Let u = ln x, du = (1/x) dx; integrand becomes ∫ 1/(1 + u^2) du.
• Result: arctan(ln x) + C.

Example E: ∫ x/(1 + x^4) dx

• Rewrite x^4 as (x^2)^2: ∫ [1/(1 + (x^2)^2)] · x dx.
• Let u = x^2, du = 2x dx; insert 2 and 1/2.
• Result: (1/2) arctan(x^2) + C.

Example F: ∫ [arcsin x]/sqrt(1 − x^2) dx

• Write as ∫ arcsin(x) · [1/sqrt(1 − x^2)] dx.
• Let u = arcsin(x), du = [1/sqrt(1 − x^2)] dx.
• Integrand becomes ∫ u du = u^2/2 + C.
• Result: (arcsin x)^2 / 2 + C.

Key Terms & Definitions

• arcsin x: Inverse sine; derivative 1/sqrt(1 − x^2) on (−1, 1).
• arctan x: Inverse tangent; derivative 1/(1 + x^2) for all real x.
• arcsec x: Inverse secant; derivative 1/(|x| sqrt(x^2 − 1)) where defined.
• Substitution: Let u = g(x), rewrite integral in u using du = g′(x) dx.

Action Items / Next Steps

• Practice identifying patterns matching arcsin, arctan, arcsec forms.
• Rehearse factoring constants to create 1 ± u^2 structures.
• Verify substitutions by differentiating final answers to the original integrands.