Question

Integrate (4x + 2/x) with respect to x.​

Answer 1

The method of integration by substitution may be used to easily compute complex integrals. Let us examine an integral of the form

ab f(g(x)) g'(x) dx

Let us make the substitution u = g(x), hence du/dx = g'(x) and du = g'(x) dx

With the above substitution, the given integral is given by

ab f(g(x)) g'(x) dx = g(a)g(b) f(u) du

Answer 2

We know that

∫√(a^2 - x^2)dx = 1\2[x√(a^2 - x^2)] + 1\2[a^2 sin(x/a)] +c

Putting the value in above, we get

∫√(4 - x^2)dx = 1/2 [x√(2^2-x^2)] +1/2 [2^2 ×sin−1(x/2)] + c

∫√(4 - x^2)dx = 1/2 [x√(4-x^2)] + 1/2 [4sin-1(x/2)] + c