Integrals formulas

All 22 formulas in this topic

Indefinite Integral∫f(x) dx=F(x)+C,F′(x)=f(x)\int f(x)\,dx = F(x) + C, \quad F'(x) = f(x)The indefinite integral finds all functions whose derivative equals a given function, expressed as an antiderivative plus an arbitrary constant.Power Rule for Integration∫xn dx=xn+1n+1+C,n≠−1\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1Power Rule for Integration instantly finds antiderivatives of polynomial terms; use it to integrate any power of x except x to the negative one.Fundamental Theorem of Calculus∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)Compute the exact value of a definite integral using the antiderivative—connect area under a curve to the accumulation of change.U-Substitution∫f(g(x))g′(x) dx=∫f(u) du\int f\big(g(x)\big)g'(x)\,dx = \int f(u)\,duU-substitution simplifies integrals by transforming them into easier forms using a change of variables, ideal when you spot a function and its derivative.Integration by Parts∫u dv=uv−∫v du\int u\,dv = uv - \int v\,duIntegration by Parts is a calculus technique for evaluating integrals of products by transforming them into simpler forms when substitution fails.Integral of 1/x∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx = \ln|x| + CIntegrating 1/x yields the natural logarithm of the absolute value of x, plus a constant—essential for reciprocal function problems.Integral of e^x∫ex dx=ex+C\int e^{x}\,dx = e^{x} + CThe integral of e^x equals e^x plus C, the essential antiderivative for solving exponential growth, decay, and differential equations.Integral of Sin x∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx = -\cos x + CThe antiderivative of sine is negative cosine plus an arbitrary constant, used to reverse differentiation of sine in calculus and physics applications.Integral of Cos x∫cos⁡x dx=sin⁡x+C\int \cos x\,dx = \sin x + CIntegrating cos x gives sin x plus a constant; use this when you need to find areas under cosine curves or solve differential equations.Trigonometric Substitutiona2−x2  ⟹  x=asin⁡θ\sqrt{a^2 - x^2} \implies x = a\sin\thetaTrigonometric substitution replaces expressions like \sqrt{a^2-x^2} with trig functions, making integrals with radicals easier to solve.Partial FractionsP(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}Partial Fractions breaks a rational function into a sum of simpler fractions, each of which is easier to integrate or analyze.Definite Integral Properties∫abf=−∫baf,∫aaf=0\int_a^b f = -\int_b^a f, \qquad \int_a^a f = 0Two key properties of definite integrals: reversing the bounds negates the integral, and integrating over a single point gives zero.Area Between CurvesA=∫ab[f(x)−g(x)] dxA = \int_a^b \big[f(x) - g(x)\big]\,dxFind the area of a region between two curves by computing the definite integral of their difference over a specified interval.Volume by DisksV=π∫ab[f(x)]2 dxV = \pi\int_a^b \big[f(x)\big]^2\,dxVolume by Disks calculates the volume of a solid of revolution by integrating the areas of circular cross-sections perpendicular to the axis of rotation.Volume by ShellsV=2π∫abx f(x) dxV = 2\pi\int_a^b x\,f(x)\,dxVolume by Shells calculates the volume of a solid of revolution around the y-axis by integrating thin cylindrical shells instead of disk slices.Arc Length of a CurveL=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + \big[f'(x)\big]^2}\,dxArc length measures the actual distance along a curve between two points, extending the simple distance formula to account for the curve's twists and turns.Average Value of a Functionfavg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dxFind the mean height of a function over an interval with the average value formula; use it for real-world average rates of change over time.Improper Integrals∫a∞f(x) dx=lim⁡t→∞∫atf(x) dx\int_a^{\infty} f(x)\,dx = \lim_{t \to \infty} \int_a^{t} f(x)\,dxEvaluate integrals over infinite intervals or involving unbounded functions by computing limits of proper integrals.Riemann Sum∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta xRiemann Sum approximates a definite integral by dividing the area under a curve into rectangles; it connects intuition to formal integration.Trapezoidal Rule∫abf(x) dx≈Δx2[f0+2f1+⋯+2fn−1+fn]\int_a^b f(x)\,dx \approx \frac{\Delta x}{2}\big[f_0 + 2f_1 + \cdots + 2f_{n-1} + f_n\big]The Trapezoidal Rule approximates definite integrals by summing trapezoid areas, useful when exact antiderivatives are difficult or impossible to find.Separable Differential Equationsdydx=g(x)h(y)  ⟹  ∫dyh(y)=∫g(x) dx\frac{dy}{dx} = g(x)h(y) \implies \int \frac{dy}{h(y)} = \int g(x)\,dxSolve differential equations where the rate of change is a product of a function of x and a function of y by separating variables and integrating.Double Integral∬Rf(x,y) dA\iint_R f(x,y)\,dACompute the total accumulation of a quantity over a 2D region—like volume under a surface or mass—by integrating a function of two variables over an area.

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