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Formulas: integral calculus, grade 11 (5 units)

Basic integrals

  • The indefinite integral – all the antiderivatives
    ∫f(x) dx=F(x)+C\displaystyle \int f(x)\,dx=F(x)+CF′(x)=f(x)\displaystyle F^{\prime}(x)=f(x)
  • A power
    ∫xn dx=xn+1n+1+C\displaystyle \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+Cn≠−1\displaystyle n\neq-1
  • A constant
    ∫k dx=kx+C\displaystyle \int k\,dx=kx+C
  • ∫1x2 dx=−1x+C\displaystyle \int\frac{1}{x^{2}}\,dx=-\frac{1}{x}+C
  • One over a root
    ∫1x dx=2x+C\displaystyle \int\frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C
  • A root
    ∫x dx=23xx+C\displaystyle \int\sqrt{x}\,dx=\frac{2}{3}x\sqrt{x}+C

The rules of integration

  • A constant factor
    ∫k⋅f(x) dx\displaystyle \int k\cdot f(x)\,dx =\displaystyle = k∫f(x) dx\displaystyle k\int f(x)\,dx
  • A sum and a difference
    ∫(f(x)±g(x))dx\displaystyle \int\left(f(x)\pm g(x)\right)dx =\displaystyle = ∫f(x) dx±∫g(x) dx\displaystyle \int f(x)\,dx\pm\int g(x)\,dx
  • A linear inner function
    ∫f(mx+b) dx=1mF(mx+b)+C\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+Cm≠0\displaystyle m\neq0
  • A composite function times the inner derivative
    ∫f(u(x))⋅u′(x) dx=F(u(x))+C\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C
  • A power of a function times its derivative
    ∫(u(x))n⋅u′(x) dx=(u(x))n+1n+1+C\displaystyle \int\left(u(x)\right)^{n}\cdot u^{\prime}(x)\,dx=\frac{\left(u(x)\right)^{n+1}}{n+1}+Cn≠−1\displaystyle n\neq-1

Trigonometric functions (in radians)

  • ∫cos⁡x dx=sin⁡x+C\displaystyle \int\cos x\,dx=\sin x+C
  • ∫sin⁡x dx=−cos⁡x+C\displaystyle \int\sin x\,dx=-\cos x+C
  • ∫1cos⁡2x dx=tan⁡x+C\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C
  • From cos⁡2x=1−2sin⁡2x\displaystyle {\cos2x}={1-2\sin^2x}
    sin⁡2x\displaystyle \sin^{2}x =\displaystyle = 1−cos⁡2x2\displaystyle \frac{1-\cos2x}{2}
  • From cos⁡2x=2cos⁡2x−1\displaystyle {\cos2x}={2\cos^2x-1}
    cos⁡2x\displaystyle \cos^{2}x =\displaystyle = 1+cos⁡2x2\displaystyle \frac{1+\cos2x}{2}
  • From sin⁡2x=2sin⁡xcos⁡x\displaystyle {\sin2x}={2\sin x\cos x}
    sin⁡xcos⁡x\displaystyle \sin x\cos x =\displaystyle = 12sin⁡2x\displaystyle \frac{1}{2}\sin2x
  • From sin⁡2x+cos⁡2x=1\displaystyle {\sin^2x+\cos^2x}={1}
    tan⁡2x\displaystyle \tan^{2}x =\displaystyle = 1cos⁡2x−1\displaystyle \frac{1}{\cos^{2}x}-1

The accumulation function and the definite integral

  • The accumulation function of f\displaystyle f from a\displaystyle a
    Fa(x)=∫axf(t) dt\displaystyle F_{a}(x)=\int_{a}^{x}f(t)\,dt
  • The fundamental theorem
    Fa′(x)=f(x)\displaystyle F_{a}^{\prime}(x)=f(x)
  • The definite integral, F\displaystyle F an antiderivative of f\displaystyle f
    ∫abf(x) dx=F(b)−F(a)\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)
  • The same limit
    ∫aaf(x) dx=0\displaystyle \int_{a}^{a}f(x)\,dx=0
  • Swapping the limits
    ∫baf(x) dx\displaystyle \int_{b}^{a}f(x)\,dx =\displaystyle = −∫abf(x) dx\displaystyle -\int_{a}^{b}f(x)\,dx
  • Joining intervals
    ∫abf(x) dx+∫bcf(x) dx\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx =\displaystyle = ∫acf(x) dx\displaystyle \int_{a}^{c}f(x)\,dx

Areas and accumulation

  • An area above the x\displaystyle x-axis
    S=∫abf(x) dx\displaystyle S=\int_{a}^{b}f(x)\,dxf(x)≥0\displaystyle f(x)\geq0
  • An area below the x\displaystyle x-axis
    S=−∫abf(x) dx\displaystyle S=-\int_{a}^{b}f(x)\,dxf(x)≤0\displaystyle f(x)\leq0
  • The area between two graphs
    S=∫ab(f(x)−g(x))dx\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dxf(x)≥g(x)\displaystyle f(x)\geq g(x)
  • The displacement from the velocity
    s(b)−s(a)=∫abv(t) dt\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt