One big advice for you, Natasha1 is that, you should open your book, and re-read the chapter that teaches you the u-substitution. read and try to understand the concept, then move on to some examples, try to understand them. And finally, you should pratice solving some integrals that involve the u-substitution.Natasha1 said:Could anyone explain to me very simply by means of a mechanical (formula) approach maybe why the integration of -x / (x^2 + 5) gives - 1/2 ln l x^2 + 5 l
Integration is a mathematical process that involves finding the area under a curve on a graph. It is the inverse operation of differentiation, and it is used to solve problems involving rates of change, motion, and accumulation.
The function -x/(x^2+5) is a rational function, which means it is a ratio of two polynomials. In this case, the numerator is -x and the denominator is x^2+5. It is a continuous function with a vertical asymptote at x=0 and a horizontal asymptote at y=0.
The antiderivative of -x/(x^2+5) is -1/2ln|x^2+5| + C, where C is the constant of integration. This can be found by using the substitution method or by recognizing the function as the derivative of -1/2ln|x^2+5|.
The integral of -x/(x^2+5) can be solved by using the substitution method, where u = x^2+5 and du = 2x dx. This will result in the integral becoming -1/2 integral of 1/u du, which is equal to -1/2ln|u| + C. Substituting back in for u, the final answer is -1/2ln|x^2+5| + C.
Yes, the integral of -x/(x^2+5) can also be solved using the partial fractions method, where the rational function is broken down into simpler fractions. It can also be solved using trigonometric substitutions or by recognizing it as a known function's derivative, such as -1/2ln|x^2+5| being the derivative of arctan(x/√5). However, the substitution method is the most straightforward and commonly used method for this integral.