Ask four questions in order. (1) Is it a basic formula after expanding or using a trigonometric identity? (2) Is one factor the derivative of something inside another? Then substitute. (3) Is it a product of two different kinds of function, such as x eˣ or x ln x? Then integrate by parts. (4) Does it contain √(a² − x²) or x² + a²? Then use a trigonometric substitution.
sin² x, cos² xIdentity: sin² x = ½(1 − cos 2x), cos² x = ½(1 + cos 2x)sin 3x cos xProduct to sum: ½(sin 4x + sin 2x)f(g(x)) g′(x)Substitution u = g(x), e.g. ∫ 2x cos(x²) dxx eˣ, x sin x, x ln x, ln xBy parts: ∫ u dv = uv − ∫ v du√(a² − x²), 1/√(a² − x²)Let x = a sin θ1/(x² + a²)Let x = a tan θEvaluate ∫02 x e⁻ˣ dx.
u = x, dv = e⁻ˣ dxdu = dx, v = −e⁻ˣ∫02 x e⁻ˣ dx = [−x e⁻ˣ]02 + ∫02 e⁻ˣ dx= −2e⁻² + [−e⁻ˣ]02 = −2e⁻² − e⁻² + 1= 1 − 3e⁻²Answer: 1 − 3e⁻² (≈ 0.594)
Find ∫ sin² x dx.
cos x factor, so substitution will not work. Use sin² x = ½(1 − cos 2x)∫ ½(1 − cos 2x) dx = ½x − ¼ sin 2x + CAnswer: ½x − ¼ sin 2x + C
Find ∫ sin² x cos x dx.
cos x is the derivative of sin x, so let u = sin x, du = cos x dx∫ u² du = u³/3 + CAnswer: (sin³ x)/3 + C
∫ sin² x dx needs an identity; ∫ sin² x cos x dx needs a substitution; ∫ x sin x dx needs parts. Decide the method from the structure, not from which chapter the question reminds you of.How to study: Click any question below to instantly load it on the coordinate grid and check your steps.