Integrate(1+x/x^3)dx
Don't know how to go bout this, if the numerator was a whole number it would ve been easier for me, Pls help me out.
(1+x/x^3)dx
= x -1/x + c
Please can you show an explanatory working, just to be able to work a similar question. thanks
To integrate the given function, you can use a method called partial fraction decomposition. The first step is to rewrite the function as a sum of simpler fractions:
1 + (x / x^3) = 1 + (1 / x^2)
Now, we can perform partial fraction decomposition by expressing the second fraction as a sum of two fractions:
1 / x^2 = A / x + B / x^2
To find the values of A and B, we can multiply the equation by x^2:
1 = A + Bx
Comparing the coefficients of x^0 and x^1, we get:
A = 1
B = 0
So, the partial fraction decomposition becomes:
1 / x^2 = 1 / x + 0 / x^2
Now, we can integrate each term separately:
∫(1+x/x^3)dx = ∫(1 + 1/x + 0)dx
= ∫1dx + ∫(1/x)dx + ∫0dx
= x + ln|x| + C
Therefore, the result of the integral is:
∫(1+x/x^3)dx = x + ln|x| + C