The antiderivative of 1/x is equal to ln |x|+C where ln is the natural logarithm (with base e) and C is a constant. In this post, we will learn how to find the antiderivative of 1/x.
The antiderivative of 1/x formula is given by
∫ dx/x = ln |x| + C.
Find the Antiderivative of 1/x
The antiderivative of 1/x is a function of x whose derivative will be 1/x. That is, the antiderivative of 1/x is the integration of 1/x with respect to x, so we need to find the integral ∫ dx/x.
Now, ∫ dx/x
Put ln x = z
Differentiating, dx/x = dz.
Hence, ∫ dx/x = ∫ dz = z+C.
Putting the value of z back, we get that
∫ dx/x = ln x +C.
As ln is defined for positive values of x, we deduce that
∫ dx/x = ln |x| +C, where C is an integral constant.
So the anti-derivative of 1/x is equal to ln |x|+C where C denotes a constant.
Now we will verify that the derivative of ln|x|+C is equal to 1/x. Note that the derivative of ln|x|+C is equal to
= 1/x +0
= 1/x, thus verified.
Q1: What is the antiderivative of 1/x?
Answer: The antiderivative of 1/x is equal to loge |x| + C where C is a constant.