Question

help me to evaluate the following integrals;

Answer 1

\[\int\limits_{}^{}(3-2y^-2)dy \]

Answer 2

can you give me the solution thanks...

Answer 3

Just apply power rule to each term.
Here is a similar example:\[\large\rm \int\limits 5+x^{-7}~dx\quad=5x+\frac{1}{-7+1}x^{-7+1}+c\]So our answer would be\[\large\rm 5x-\frac{x^{-6}}{6}+c\]

Answer 4

What are you having trouble with? :o

Answer 5

can you do that problem for me, so that i can understand...

Answer 6

Some Rules:

\(\color{#000000}{\displaystyle\int\limits_{~}^{~} x^n~dx=\frac{x^{n+1}}{n+1}\color{grey}{\rm +C}}\) (for all besides n=-1)

\(\color{#000000}{\displaystyle\int\limits_{~}^{~} x^{-1}~dx=\ln|x|\color{grey}{\rm +C}}\)


Also, whenever you know that,
\(\color{#000000}{\displaystyle\int\limits_{~}^{~} f(x)~dx=g(x){\rm +C}}\)
Then,
\(\color{#000000}{\displaystyle\int\limits_{~}^{~} a\cdot f(x)~dx=a\cdot g(x){\rm +C}}\)
\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, red,border:2px solid ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\(\tiny\\[1.0em]\)
Some examples:


\(\color{#000000 }{ \displaystyle \int \left(4-z^{-3}\right)~dz =4z-\frac{z^{-3+1}}{-3+1}\color{grey}{\rm +C} \\[1.5em] \displaystyle =4z-\frac{z^{-2}}{-2} \color{grey}{\rm +C}=4z+\frac{z^{-2}}{2}\color{grey}{\rm +C}}\)
\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\(\tiny\\[1.0em]\)
\(\color{#000000 }{ \displaystyle \int \left(w-8w^{4}\right)~dz = \int \left(w^1-8w^{4}\right)~dz=\frac{w^{1+1}}{1+1}-8\frac{w^{4+1}}{4+1}\color{grey}{\rm +C}}\)\(\tiny\\[1.5em]\)
\(\color{#000000 }{ \displaystyle =\frac{w^{2}}{2}-8\frac{w^{5}}{5}\color{grey}{\rm +C}=\frac{1}{2}w^{2}-\frac{8}{5}w^{5}\color{grey}{\rm +C}}\)
\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\(\tiny\\[1.0em]\)
\(\color{#000000 }{ \displaystyle \int \left(t^2+t^3+t^{-1}\right)~dz =\frac{t^{2+1}}{2+1}+\frac{t^{3+1}}{3+1}+\ln|t|\color{grey}{\rm +C}}\)\(\tiny\\[1.5em]\)
\(\color{#000000 }{ \displaystyle =\frac{t^{2}}{2}+\frac{t^{3}}{3}+\ln|t|\color{grey}{\rm +C}}\)