Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 3.  First, we use our inverse power rule to find the antiderivative. So, we have that $\displaystyle \int x^ndx=\frac{x^{n+1}}{n+1}$.  Once you find the antiderivative, we must remember that  where $\displaystyle F$ is the indefinite integral.  So, we plug in our limits and subtract the two.  So, we have $\displaystyle (3x^2)|_1^3=3(3^2)-3(1^2)=24$.

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 4.  First, we use our inverse power rule to find the antiderivative. So since $\displaystyle x$ is to the power of $\displaystyle -1$, we have that $\displaystyle \int x^{-1}=ln|x|$.  Once you find the antiderivative, we must remember that  where $\displaystyle F$ is the indefinite integral.  So, we plug in our limits and subtract the two.  So, we have $\displaystyle ln(x)|_1^4=ln(4)-ln(1)=ln(4)$ because we know that $\displaystyle ln\ 1=0$.

First, we know that we can pull the constant "4" out of the integral, and we then evaluate the integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

First, we know that the integral of a sum is the same as the sum of the integrals, so if needed, we can split the three integrals up and evaluate them seperately.  We then evaluate each integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

Normally, we would evalute the indefinite integral according to the following equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. However, in this case, $\displaystyle n=-1$.  Now we use our other rule that states the integral of $\displaystyle \frac{1}{x}$ is equal to $\displaystyle \ln (x)$ plus a constant.  From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

We evaluate the integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above. Keep in mind that $\displaystyle \frac{1}{x^{2}}$ is the same as $\displaystyle x^{-2}$.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

First, we remember that the integral of a sum is the same as the sum of the integrals, so we can split the sum into seperate integrals and solve them individually.  We then evaluate each integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

First, we know that we can pull the constant $\displaystyle \frac{1}{2}$ out of the integral, and we then evaluate the integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

First, we know that we can pull the constant $\displaystyle \frac{3}{5}$ out of the integral, and we then evaluate the integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.

We evaluate the integral according to this equation:

$\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$. Keep in mind that $\displaystyle \sqrt{x}$ is the same as $\displaystyle x^{1/2}$. From this, we acquire the answer above.  As a note, we cannot forget the constant of integration $\displaystyle C$ which would be lost during the differentiation.