# Question Video: Finding the General Antiderivative of a Function Mathematics • Higher Education

What is the antiderivative πΉ of π(π₯) = β5 + (1 + π₯Β²)β»ΒΉ that satisfies πΉ(1) = 0?

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### Video Transcript

What is the antiderivative capital πΉ of π of π₯ equals negative five plus one plus π₯ squared to the power of negative one that satisfies capital πΉ of one equals zero?

Here, weβve been given a rather nasty looking function and asked to work out its antiderivative. That is, the function that, when differentiated, gives us negative five plus one plus π₯ squared to the power of negative one. Now, if we rewrite one plus π₯ squared to the power of negative one as one over one plus π₯ squared, we can quote the general result for the derivative of the inverse tan of ππ₯. Itβs π over one plus π of π₯ all squared for real constants π. Then, if we let π be equal to one, we find that the derivative of the inverse tan of π₯ is one over one plus π₯ squared. Well, we actually have that expression in our function. And that tells us that the antiderivative of one over one plus π₯ squared must be the inverse tan of π₯. So what about the antiderivative of negative five?

Well, the antiderivative of negative five is negative five π₯. So we find that an antiderivative capital πΉ could be negative five π₯ plus inverse tan of π₯. Remember though, we need to include some constant. And this is because when we differentiate a constant, we end up with zero. Luckily, we can work out the value of this constant by using the fact that capital πΉ of one is equal to zero. Letβs substitute π₯ equals one into our expression for the antiderivative. When we do, we obtain zero to be equal to negative five times one plus the inverse tan of one plus π. Negative five times one is negative five. And the inverse tan of one is π by four. We solve this equation by adding five to both sides and then subtracting π by four. And we see that π is equal to five minus π by four.

And so we see the antiderivative capital πΉ is defined by the function negative five π₯ plus the inverse tan of π₯ minus π by four plus five.