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 ππ₯ plus our constant of integration
πΆ. 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 canβt 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
πΆ.
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 πΆ.