First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{3x^3}{3}-\frac{x^2}{2}+x=x^3-\frac{x^2}{2}+x\)

Now, evaluate at 3 and then 1. Subtract the results:

\(\displaystyle (27-\frac{9}{2}+3)-(1-\frac{1}{2}+1)=28-4=24\)

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{x^{\frac{5}{2}}}{\frac{5}{2}}\)

Simplify:

\(\displaystyle \frac{2}{5}x^{\frac{5}{2}}\)

Now evaluate at 2 and then 0. Subtract the results:

\(\displaystyle \frac{2}{5}(2^{\frac{5}{2}})-\frac{2}{5}(o^{\frac{5}{2}})=2.2627-0=2.2627\)

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{3x^3}{3}+\frac{2x^2}{2}+2x\)

Simplify:

\(\displaystyle x^3+x^2+2x\)

Evaluate at 3 and then at 1. Subtract the results:

\(\displaystyle (27+9+6)-(1+1+2)=42-4=38\)

First, integrate. Remember to add one to the exponent and also put that result on the denominator:

\(\displaystyle \frac{x^3}{3}-\frac{x^2}{2}+4x\)

Now, evaluate at 4 and then 1. Subtract the results:

\(\displaystyle (\frac{64}{3}-8+16)-(\frac{1}{3}-\frac{1}{2}+4)\)

Simplify to get your answer:

\(\displaystyle 4+\frac{129}{6}=\frac{51}{2}\)

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{x^5}{5}+\frac{x^2}{2}-x\)

Evaluate at 2 and then 0. Subtract the results:

\(\displaystyle (\frac{32}{5}+2-2)-0=\frac{32}{5}\)

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{4x^3}{3}-3x\)

Now, evaluate at 2 and then at 1. Subtract the results:

\(\displaystyle (\frac{32}{3}-6)-(\frac{4}{3}-3)\)

Simplify to get your answer:

\(\displaystyle \frac{28}{3}-3=\frac{19}{3}\)

Step 1: Integrate:

\(\displaystyle \int_{1}^{3}3x^4+7x+3\rightarrow \frac {3x^5}{5}+\frac {7x^2}{2}+3x|_{1}^{3}\)

Step 2: Evaluate at the upper limit:

Plug in \(\displaystyle x=3\).

\(\displaystyle \frac {3(3)^5}{5}+\frac {7(3)^2}{2}+3(3)=\frac {3^6}{5}+\frac {63}{2}+9=186.3\)

Step 3: Evaluate at the lower limit:

Plug in \(\displaystyle x=1\).

\(\displaystyle \frac {3(1)^5}{5}+\frac {7(1)^2}{2}+3(1)=\frac {3}{5}+\frac {7}{2}+3=7.1\)

Step 4: Take the valuation at the lower limit and subtract it FROM the upper limit:

\(\displaystyle 186.3-7.1=179.2=\frac {896}{5}\)

The integration of \(\displaystyle 3x^4+7x+3\) is \(\displaystyle \frac {896}{5}\)

Step 1: Take the antiderivative of each term:

\(\displaystyle 2x^3 \rightarrow 2 \cdot \frac {x^4}{4}=\frac {x^4}{2}\)

\(\displaystyle 3x^2=3 \cdot \frac {x^3}{3}=x^3\)

\(\displaystyle 3x\rightarrow 3 \cdot \frac {x^2}{2}=\frac {3x^2}{2}\)

Step 2: Put all the antiderivatives in step 1 together based on the signs in the integral...

\(\displaystyle \frac {x^4}{2}+x^3+\frac {3x^2}{2}\)

Step 3: Plug in the upper and lower limits:

Upper Limit is \(\displaystyle 2\), lower limit is \(\displaystyle 0\).

Plug in \(\displaystyle x=2\):

\(\displaystyle \frac {2^4}{2}+2^3+\frac {3(2)^2}{2}=2^3+2^3+3(2)=8+8+6=22\)

Plug in \(\displaystyle x=0\). Since all terms have \(\displaystyle x\), the value will be \(\displaystyle 0\)

Step 4: Subtract the value of the lower limit from the upper limit:

\(\displaystyle 22-0=22\)

The value of this integral is \(\displaystyle 22\).

First, integrate this expression. Remember to raise the exponent by 1 and then also put that result on the denominator:

\(\displaystyle \frac{4x^3}{3}-\frac{x^2}{2}\)

Now, evaluate at 2 and then 0. Subtract the results:

\(\displaystyle (\frac{32}{3}-2)-(0)\)

Simplify to get your answer:

\(\displaystyle \frac{32}{3}-\frac{6}{3}=\frac{26}{3}\)

First, take the 4 outside of the integral sign and rewrite the radical as a fractional exponent. It's easier to visualize that when integrating:

\(\displaystyle 4\int x^{\frac{1}{2}}dx\)

Now integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

\(\displaystyle \frac{x^{\frac{3}{2}}}{\frac{3}{2}}\)

Simplify and multiply by the 4 that you took out:

\(\displaystyle 4(\frac{2}{3})x^{\frac{3}{2}}=\frac{8}{3}x^{\frac{3}{2}}\)

Now evaluate at 2 and then 0. Subtract the results:

\(\displaystyle \frac{8}{3}2^{\frac{3}{2}}-\frac{8}{3}0^{\frac{3}{2}}=7.5425\)