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Integrate

Integral (SymPy, WMA)

Integrate[f, x]

integrates f with respect to x. The result does not contain the additive integration constant.

Integrate[f, {x, a, b}]

computes the definite integral of f with respect to x from a to b.

Integrate a polynomial:

• • Integrate[6 x ^ 2 + 3 x ^ 2 - 4 x + 10, x]

Integrate trigonometric functions:

• • Integrate[Sin[x] ^ 5, x]

Definite integrals:

• • Integrate[x ^ 2 + x, {x, 1, 3}]

• • Integrate[Sin[x], {x, 0, Pi/2}]

Some other integrals:

• • Integrate[1 / (1 - 4 x + x^2), x]

• • Integrate[4 Sin[x] Cos[x], x]

• • Integrate[-Infinity, {x, 0, Infinity}]

Integrating something ill-defined returns the expression untouched:

• • Integrate[1, {x, Infinity, 0}]

Here how is an example of converting integral equation to TeX:

• • Integrate[f[x], {x, a, b}] // TeXForm

Sometimes there is a loss of precision during integration.
You can check the precision of your result with the following sequence of commands.

• • Integrate[Abs[Sin[phi]], {phi, 0, 2Pi}] // N

• • % // Precision

• • Integrate[ArcSin[x / 3], x]

• • Integrate[f'[x], {x, a, b}]

and,

• • D[Integrate[f[u, x],{u, a[x], b[x]}], x]

  • $${\int }_{a\left[x\right]}^{b\left[x\right]}{f}^{(0,1)}\left[u,x\right]\text{𝑑}u+f\left[b\left[x\right],x\right] b^{\prime }\left[x\right]\mathrm{-}f\left[a\left[x\right],x\right] a^{\prime }\left[x\right]$$

• • N[Integrate[Sin[Exp[-x^2 /2 ]],{x,1,2}]]

  • $$0.330804$$