[Solved] What is the Antiderivative of 1/x

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.

VERIFICATION:

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

d/dx(ln|x|+C)

= 1/x +0

= 1/x, thus verified.

ALSO READ:

Antiderivative of 1

Antiderivative of x

Antiderivative of 1/x

FAQs