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

Video Transcript

Determine the integral of five over two plus four over 𝑥 d𝑥.

In this question, we are asked to integrate the expression five over two plus four over 𝑥 with respect to 𝑥. 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 𝑘 with respect to 𝑥 is equal to 𝑘𝑥 plus our constant of integration 𝐶. This means that integrating five over two gives us five over two 𝑥 or five 𝑥 over two. And we know that we can just add one constant of integration at the end.

Integrating four over 𝑥 is more complicated, as we can’t do this by using the power rule for integration because our exponent of 𝑥 is negative one. Instead, we recall the rule for integrating a reciprocal function 𝑎 over 𝑥 with respect to 𝑥. This is equal to 𝑎 multiplied by the natural logarithm of the absolute value of 𝑥 plus the constant of integration 𝐶.

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