Integration techniques

U-substitution practice problems \int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x) 12 u-substitution practice problems with answers and full step-by-step solutions, from (x²+1)⁵ to ln x / x, back-substitution and definite integrals. Integration by parts practice problems \int u\,dv = uv - \int v\,du 12 integration by parts practice problems with answers and full solutions: x eˣ, ln x, arctan x, repeated parts and the e^x sin x loop. Trig integral practice problems \sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2} 12 trigonometric integral practice problems with answers: powers of sine and cosine, tangent and secant, half-angle and product-to-sum. Trig substitution practice problems \sqrt{a^2 - x^2}:\ x = a\sin\theta \quad \sqrt{a^2 + x^2}:\ x = a\tan\theta \quad \sqrt{x^2 - a^2}:\ x = a\sec\theta 12 trigonometric substitution practice problems with answers and full solutions for √(a²−x²), √(a²+x²), √(x²−a²) and completing the square. Partial fractions practice problems \frac{px + q}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b} 12 partial fraction decomposition practice problems with answers: distinct and repeated factors, irreducible quadratics and long division first.