integral of sin^3 x / cos^4 x dx

integral of sin^3 x / cos^4 x dx how to integrate sin^3 x / cos^4 x ∫ sin^3 x / cos^4 x dx step by step rewrite sin^3 x as sin^2 x * sin x sin^2 x = 1 - cos^2 x integral becomes ∫ (1 - cos^2 x) sin x / cos^4 x dx substitution u = cos x, du = - sin x dx integral becomes -∫ (1 - u^2) / u^4 du simplify to -∫ (u^(-4) - u^(-2)) du integrate term by term: -[ -1/(3u^3) + 1/u ] + C final answer: 1/(3 cos^3 x) - 1/cos x + C or sec^3 x/3 - sec x + C alternate: write as ∫ tan^3 x sec x dx tan^3 x sec x = tan^2 x * tan x sec x = (sec^2 x - 1) tan x sec x then substitute u = sec x, du = sec x tan x dx technique for odd power of sine in numerator antiderivative of sin^3 x / cos^4 x example of trig integral with sine odd power simplify and use u-substitution method integral of powers of sine over powers of cosine step-by-step u-substitution with u = cos x ∫ sin^3 x / cos^4 x dx = (1/3) sec^3 x - sec x + C detailed solution using substitution rewrite in terms of secant and tangent Calculus syllabus American calculus curriculum Canadian calculus curriculum AP Calculus AB AP Calculus BC Precalculus Differential calculus Integral calculus Limits and continuity Derivatives rules Chain rule Product rule Quotient rule Implicit differentiation Optimization problems Related rates problems Riemann sums Definite integrals Indefinite integrals Fundamental Theorem of Calculus u-substitution Integration by parts Graphing derivatives Applications of derivatives Applications of integrals Polar coordinates calculus Series and sequences Convergence tests Maclaurin series Taylor series Multivariable calculus Partial derivatives Double integrals Triple integrals Vector calculus    • Integration by substitution      • Integration by parts      • Trigonometric Integrals      • THE INTEGRATION OF EXPONENTIAL FUNCTIONS      • integration by partial fractions      • definite integrals   Your go-to hub for Calculus Made Easy — from calculus basics to advanced topics. Whether you're looking for clear explanations, step-by-step Integral ForYou lessons, or deep dives with Integrals Education and INTEGRALS TV, we’ve got you covered. Even if you're tackling Química Integrals, this channel simplifies everything to boost your understanding and grades! Music: Local Elevator by Kevin MacLeod Free music licensed under Creative Commons: Attribution 4.0 License https://creativecommons.org/licenses/... Source: https://incompetech.com