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

Determine ∫ (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 𝐢. 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 can’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‘Ž over π‘₯ with respect to π‘₯. This is equal to π‘Ž multiplied by the natural logarithm of the absolute value of π‘₯ plus the constant of integration 𝐢.

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 𝐢.

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