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.
