# Mr. Nasty ## Only has trig functions - Product of sines and cosines - Odd cosine power - Save one cosine factor and use $\cos^2{x} = 1 - \sin^2{x}$ to express the rest in terms of $\sin{x}$ and substitute $u = \sin{x}$: $\int{\sin^m{x}\cos^{2k+1}{x}}dx = \int{\sin^m{x}(\cos^2{x})^k\cos{x}dx} = \int{\sin^m{x}(1 - \sin^2{x})^k\cos{x}dx}$ - Odd sine power - Save one sine factor and use $\sin^2{x} = 1 - \cos^2{x}$ to express the rest in terms of $\cos{x}$ and substitute $u = \cos{x}$: $\int{\cos^m{x}\sin^{2k+1}{x}}dx = \int{\cos^m{x}(\sin^2{x})^k\sin{x}dx} = \int{\cos^m{x}(1 - \cos^2{x})^k\sin{x}dx}$ - Both even powers - Use the identities $\sin^2{x} = \frac{1}{2}(1 - \cos{2x})$ and $\cos^2{x} = \frac{1}{2}(1 + \cos{2x})$ to express everything in terms of $\cos{2x}$ and restart the flowchart. - Same powers - $\int{(\sin{x}\cos{x})^n dx}$ - Use the identity $\sin{x}\cos{x} = \frac{1}{2}\sin{2x}$ to express everything in terms of $\sin{2x}$ and restart the flowchart. - $\int{\sin{mx}\cos{nx}}dx$ - $\sin{a}\cos{b} = \frac{1}{2}(\sin{(a - b)} + \sin{(a + b)})$ - $\int{\sin{mx}\sin{nx}dx}$ - $\sin{a}\sin{b} = \frac{1}{2}(\cos{(a - b)} - \cos{(a + b)})$ - $\int{\cos{mx}\cos{nx}dx}$ - $\cos{a}\cos{b} = \frac{1}{2}(\cos(a - b) + \cos(a + b))$ - Product of secants and tangents - Even secant power $\ge 2$ - Save a factor of $\sec^2{x}$ and use $\sec^2{x} = 1 + \tan^2{x}$ to express everything else in terms of $\tan{x}$ and substitute $u = \tan{x}$: $\int{\tan^m{x} \sec^{2k}{x} dx} = \int{\tan^m{x} (\sec^2{x})^{k - 1} \sec^2{x} dx} = \int{\tan^m{x} (1 + \tan^2{x})^{k - 1} \sec^2{x} dx}$ - Odd tangent power - Save a factor of $\sec{x}\tan{x}$ and use $\tan^2{x} = \sec^2{x} - 1$ to express everything else in terms of $\sec{x}$ and substitute $u = \sec{x}$: $\int{\tan^{2k + 1}{x} \sec^n{x} dx} = \int{(\tan^2{x})^k \sec^{n - 1}{x} \sec{x} \tan{x} dx} = \int{(\sec^2{x} - 1)^k \sec^{n - 1}{x} \sec{x} \tan{x} dx}$ - Use anti-derivatives of $\sec{x}$ and $\tan{x}$ - $\int{\sec{x} dx} = \ln{|\sec{x} + \tan{x}|} + C$ - $\int{\tan{x} dx} = \ln{|\sec{x}| + C}$ ## Rational function - $\frac{du}{u}$ - substitution - Factorable denominator - Degree top $\ge$ degree bottom - Long division - Partial Fraction - Distinct linear factors - $\frac{A_1}{a_1x + b_1} + \frac{A_2}{a_2x + b_2} \cdots$ - Repeated linear factors - $\frac{a}{(ax + b)^n} = \frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \cdots + \frac{A_n}{(ax + b)^n}$ - Distinct irreducible quadratic factors - $\frac{a}{(x^2 + a_1^2)(x^2 + a_2^2)} = \frac{Ax + B}{x^2 + a_1^2} + \frac{Cx + D}{x^2 + a_2^2}$ - Complete the square if necessary - Use $\arctan{x}$ - Repeated irreducible quadratic factor - $\frac{n}{(ax^2 + bx + c)^n} = \frac{A_1 + B_1}{ax^2 + bx + c} + \frac{A_2 + B_2}{(ax^2 + bx + c)^2} + \cdots + \frac{A_n + B_n}{(ax^2 + bx + c)^n}$ - Degree top $\lt$ degree bottom - Partial Fraction - Unfactorable denominator - $x^2 + a^2$ term - Trig substitution with $x = b\tan{\theta}$ - Complete the square to create an $x^2 + a^2$ term - Trig substitution with $x = b\tan{\theta}$ ## Radical expression - $\sqrt{a^2 - x^2}$ - Substitute $x = a\sin{\theta}$ - $\sqrt{a^2 + x^2}$ - Substitute $x = a\tan{\theta}$ - $\sqrt{x^2 - a^2}$ - Substitute $x = a\sec{\theta}$ ## By parts friendly factors - $\mathrm{polynomial \times transcendental(trig, exp, log)}$ - Integration by parts