Use the inverse Power Rule to evaluate the integral.  We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$.  We see that this rule tells us to increase the power of $\displaystyle x$ by 1 and multiply by $\displaystyle \frac{1}{n+1}$.  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  Firstly, constants can be taken out of integrals, so we pull the 3 out front.  Next, according to the inverse power rule, we know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$.  We see that this rule tells us to increase the power of $\displaystyle x$ by 1 and multiply by $\displaystyle \frac{1}{n+1}$. Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$.  We see that this rule tells us to increase the power of $\displaystyle x$ by 1 and multiply by $\displaystyle \frac{1}{n+1}$.  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$.  We see that this rule tells us to increase the power of $\displaystyle x$ by 1 and multiply by $\displaystyle \frac{1}{n+1}$  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  Firstly, constants can be taken out of the integral, so we pull the 1/2 out front and then complete the integration according to the rule. We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$.  We see that this rule tells us to increase the power of $\displaystyle x$ by 1 and multiply by $\displaystyle \frac{1}{n+1}$.  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$. But, in this case, $\displaystyle n$ IS equal to $\displaystyle -1$ so a special condition of the rule applies.  We must instead use $\displaystyle \int x^{-1}dx=ln|x|+C$.  Evaluate accordingly.  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Use the inverse Power Rule to evaluate the integral.  We know that $\displaystyle \int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $\displaystyle n \ne -1$. But, in this case, $\displaystyle n$ IS equal to $\displaystyle -1$ so a special condition of the rule applies.  We must instead use $\displaystyle \int x^{-1}dx=ln|x|+C$.  Pull the constant "3" out front and evaluate accordingly.  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 0 to 2.  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 $\displaystyle (\frac{1}{3}x^3+3x)|_0^2=[\frac{1}{3}2^3+3(2)]-[\frac{1}{3}0^3+3(0)]=26/3$.

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 $\displaystyle ln\ 1=0$.