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Integrate mark understanding

in this video we're going to go over some basic integration rules so let's say if you want to integrate dx for example this will give you x plus some constant c if you wish to integrate d y this will give you y plus c or if you want to integrate dt that will give you t plus c and so this is the variable of integration and make sure you always add that particular variable based on what you see here now what if we have a constant so let's say if we wish to integrate a constant with a variable of x this is going to equal kx plus c so if i want to integrate 5 dx this is going to be 5x plus c if i wish to integrate let's say 12 d y that's going to be 12y plus c or 4 dt this is going to be 4t plus c now another rule that you need to be familiar with is the power rule the anti-derivative of x raised to the n dx is equal to x raised to the n plus one divided by n plus one plus some constant c so let me give you some examples of using this power rule so let's say if we wish to find the anti-derivative of x squared dx this is going to be x raised to the 2 plus 1 divided by 2 plus 1 plus c so it's x cubed over 3 plus c which you could write that as 1 3 x cubed plus c so let's try some more examples using the power rule go ahead and find the anti-derivative of x cubed 10 x to the fourth and also let's say one over x squared go ahead and try these problems so for this one 3 plus 1 is 4 so it's going to be x to the fourth over 4 plus c for the next one 4 plus 1 is 5 so it's x to the fifth over 5. but we could simplify it 10 divided by 5 is 2 so it's 2x to the fifth power plus c now what about the last one 1 over x squared what can we do for that example before you integrate it you need to rewrite it so we can rewrite it as x raised to the negative 2 dx and then use the power rule so it's x raised to negative two plus one divided by negative two plus one plus c and so negative two plus one is negative one and then we can rewrite the final answer as negative one over x plus c so anytime you have a rational function like this one make sure to rewrite it so let's try another simple example let's say if we want to integrate 7 over x to the fourth power try that one so first let's rewrite it as seven x raised to the negative four and then...