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In order to mathematicsintegration of quadratic function, we know the general quadratic equation is ax^2 + 2bx + c = 0, this is a quadratic integral. Let we see the integration when a quadratic function in the denominator. Integration of 1 / (ax^2 + 2bx + c) with respect to x. In order to integrate given integral function first we take common a and get 1/(ax^2 + 2bx + c) = 1/a (1 / (x^2 + 2b/a x + c/a)), now we try to make a perfect square of x^2 + 2b/a x + c/a = (x + b/a) ^2 + (square root [(Ac – b^2) /a]) ^2, Now we take integration both side and get Integration of 1/(ax^2 + 2bx + c) = Integration 1/a [(x + b/a) ^2 + (square root [(Ac – b^2) /a]) ^2], we take 1/a out of integration. Integration of 1/(ax^2 + 2bx + c) =1/a Integration [(x + b/a) ^2 + (square root [(Ac – b^2) /a]) ^2], Now we use formula integration of 1/(x^2 + a^2) = 1/a tan^-1 (x/a). è Integration of 1/ (ax^2 + 2bx + c) = 1/square root (ac – b^2) tan^-1 (ax +b) /square root (AC – b^2).
Let us see some examples of integration of quadratic integral. Example: - Find the integration of the given fraction 1/ (x^2 + 4x + 13). Solution: - In order to integrate given function first we make a perfect square of giving quadratic equation x^2 + 4x + 13 = (x + 2) ^2 + 3^2. Now we can write as Integration 1/ (x^2 + 4x + 13) dx = integration 1/[(x + 2) ^2 + 3^2] dx. Now use formula integration of 1/(x^2 + a^2) = 1/a tan^-1 (x/a), now we get Integration 1/ (x^2 + 4x + 13) dx = 1/3 tan^-1 [(x + 2) / 3]. In this type we can integrate quadratic functions.
These examples are quadratic not repeated factor. We have some other formulas of quadratic integration. (1) Integration of 1/square root (a^2 – x^2) = sin^-1 (x / a), (2) integration of 1/square root (a^2 + x^2) = sinh^-1 (x/a) = log (x + square root (x^2 + a^2)) /a. (3) Integration of 1 / square toot (x^2 – a^2) = cos h^-1 (x / a) = log (x + square root (x^2 - a^2)) /a. (4) Integration of square root (a^2 – x^2) dx = x square root (a^2 – x^2) / 2 + a^2 / 2 sin^-1 (x/a). These formulas are very important formulas to integrate the quadratic functions. Let we see another example of integration of a quadratic function. Example: - Integral of 1 / square root (4 – x^2). Solution: - In order integrate given function we use the formula or substitution method. If we want to use the formula Integration of 1/square root (a^2 – x^2) = sin^-1 (x / a). We write integration of the 1 / square root (4 – x^2) = integration of 1 / square root (2^2 – x^2) = sin^-1 (x/a). This is how the quadratic integral of any quadratic function is calculated.
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