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Constant factor rule of integration is similar to the constant factor rule of derivative.Integral of f (x) = F (x) + c is the general way of writing help integration of a function f(x).For example: integral of x = (x^2)/2 + c. Here F (x) = (x^2) /2 and ‘c’ is the constant of integration.Now let us see what would be the integral of a function which is associated with a constant factor.
Integral of k * f(x).dx = k * integral of f(x).dx = k * F(x) + c
Proof: We know that, y = integral of dy. It implies that y = integral of (dy/dx).dx. Multiplying both sides by constant k we get,k * y = k * integral of (dy/dx).dx. Constant factor rule of derivative states that : d/dx (k * y) = k * dy/dx. Integratingequation 2 we get,
Integral of d/dx(k * y)dx = integral of (k * dy/dx).dx
k * y = integral of (k * dy/dx).dx …..(3)
from equation 1 and 3 we get,
integral of (k * dy/dx).dx = k * integral of (dy/dx).dx
Let us assume dy/dx = f (x)
integral of k * f(x).dx = k * integral of f(x).dx
Hence proved.
Consider an example: y = 4/x
Integral of y = integral of 4/x= integral of (4 * 1/x) (here 4 is a constant factor associated with the function 1/x). So this becomes 4 * integral of (1/x) which is equal to 4 * ln x + c. So Integral of a/x isaln x + c.
Constant factor rule and sum rule:
Integral of [k * (f (x) + g (x)] = k * integral of f(x) + k * integral of g(x)
Integral of [k * (f (x) - g (x)] = k * integral of f (x) - k * integral of g (x)
Constant factor rule and by parts rule:
Integral of [k * (f (x) * g (x)] = k * [integral of f (x) * g (x)]
Integral of f (x) *g (x) can be solved using by part’s rule.
Solved examples:
Y = 4 * integral of 9x^2).dx
Y = 4 * x^3/3 + c
= (4/3)x^3 + c
Y = (2/3) * integral of (1/x).dx + (2/3) * integral of (sin x).dx
= (2/3) * ln x + (2/3) * (-cos x) + c
= (2/3) * ln x – (2/3) * cos x + c
Y = 5 * integral of ln x + c
Use by parts rule to find an integration of ln x.
Consider ln x = 1 * ln x f (x) = 1 and g (x) = ln x
Integrating by parts we get, integral of ln x = x * ln x – x + c
Hence, Y = 5 * (ln x – x + c)
= 5*x.ln x – 5x + c’
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