Step 1: Rewrite the problem.

integral (x³+x^1/2) dx from 0 to 1 

Step 2: Integrate 

x⁴/4 + 2x^(2/3)/3 from 0 to 1

Step 3: Plug in bounds and solve. 

[1⁴/4 + 2(1)^(2/3)/3] – [0⁴/4 + 2(0)^(2/3)/3] = (1/4) + (2/3) = (3/12) + (8/12) = 11/12

Integral from 1 to 2 of (1/x³) dx

Integral from 1 to 2 of (x^-3) dx

Integrate the integral. 

 from 1 to 2 of (x^–2/-2)  

(2^–2/–2) – (1^–2/–2) = (–1/8) – (–1/2)=(3/8)

In order to evaluate the indefinite integral, ask yourself, "what expression do I differentiate to get 4".  Next, use the power rule and increase the power of \(\displaystyle x\) by 1. To start, we have \(\displaystyle 4x^{0}\), so in the answer we have \(\displaystyle 4x^{1}\).  Next add a constant that would be lost in the differentiation.  To check your work, differentiate your answer and see that it matches "4".

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 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.