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

Determine β« (5/2) + (4/π₯) dπ₯.

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### Video Transcript

Determine the integral of five over two plus four over π₯ dπ₯.

In this question, we are asked to integrate the expression five over two plus four over π₯ with respect to π₯. And we can do this by integrating term by term.

The first term of our expression is a constant. And we recall that integrating any constant π with respect to π₯ is equal to ππ₯ plus our constant of integration πΆ. This means that integrating five over two gives us five over two π₯ or five π₯ over two. And we know that we can just add one constant of integration at the end.

Integrating four over π₯ is more complicated, as we canβt do this by using the power rule for integration because our exponent of π₯ is negative one. Instead, we recall the rule for integrating a reciprocal function π over π₯ with respect to π₯. This is equal to π multiplied by the natural logarithm of the absolute value of π₯ plus the constant of integration πΆ.

Integrating four over π₯ is therefore equal to four multiplied by the natural logarithm of the absolute value of π₯. Adding the constant of integration at the end, we now have our final answer. The integral of five over two plus four over π₯ with respect to π₯ is five π₯ over two plus four multiplied by the natural logarithm of the absolute value of π₯ plus πΆ.

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