Computing $\int_0^1 \frac{1 + 3x +5x^3}{\sqrt{x}}\ dx$

$$\int_0^1 \frac{1 + 3x +5x^3}{\sqrt{x}}\ dx$$

My idea for this was to break each numerator into its own fraction as follows

$$\int_0^1 \left(\frac{1}{\sqrt{x}} + \frac{3x}{\sqrt{x}} + \frac{5x^3}{\sqrt{x}}\right)dx$$

$$\int_0^1 (x^{-1/2} + 3x^{1/2} +5x^{5/2})\ dx$$

$$\int_0^1 2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2}$$

Not really sure where to go from there. Should I sub 1 in for the x values and let that be the answer?

• You seem to have miswritten the last step. $\int_0^1 (x^{-1/2} + 3x^{1/2} +5x^{5/2}) dx = 2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2}|_{x=0}^1$ – WaveX Mar 24 '17 at 16:13

It looks like you've already done the integration correctly, but forgot to take off the integration sign. The last line should be

$$\left[2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2} \right]_0^1$$

which you can evaluate by substituting $x=1$ and $x=0$, and subtracting.

You asked whether you should substitute $x=1$: this will happen to give you the right answer, but only coincidentally, because the expression becomes $0$ when $x=0$.

Hint: the formula $$\int x^c\, dx = \frac{x^{c+1}}{c+1} + C$$ applies to all real numbers $c \neq -1$, not just integers.

• Ok.. Not sure what you are saying? Should I apply this formula to my final step? Or perhaps to the formula before I integrate it ? – John Allen Mar 24 '17 at 16:11
• The first term in the integral is $\int_0^1 2x^{1/2}\, dx = \left. \frac{2}{3/2} x^{3/2}\right|_{x=0}^{x=1} = \frac{4}{3}$. You can apply this formula to the other terms as well. – Connor Harris Mar 24 '17 at 16:14
• Yes I see now. Thanks very much! – John Allen Mar 24 '17 at 16:15
• @ConnorHarris But that would be integrating twice. The integration has already been done; the mistake was not to remove the integral sign. – Théophile Mar 24 '17 at 16:18
• @Théophile Ooh, you're right. I guess John Allen's "sub in $x=1$" is correct (properly speaking, he wants the difference between $x=1$ and $x=0$, but the expression has value $0$ for $x=0$). – Connor Harris Mar 24 '17 at 16:22

In derivative we subtract 1.

In integration we add 1.

Your solution is fine. Except last step.

$$=\int_0^1 (x^{-1/2} + 3x^{1/2} +5x^{5/2}) dx$$

$$=\left[2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2}\right]_0^1$$

$$=\left[2\cdot1^{1/2} + 2\cdot1^{3/2} + \frac{10}{7}\cdot1^{7/2}\right]-\left[2\cdot0^{1/2} + 2\cdot0^{3/2} + \frac{10}{7}\cdot0^{7/2}\right]$$

Hope now you can proceed.

• $$\int x^{\color{red}{n}}~dx=\frac{x^{n+1}}{n+1}+\color{red}{C}$$ – projectilemotion Mar 24 '17 at 16:25

Putting $t=\sqrt{x}$ you have $dt=\frac{1}{2\sqrt{x}}dx$ and the limits stay the same. $$\int_0^12(1+3t^2+5t^6)$$

Your work is correct, but for the last step write: $$\left[ 2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2}\right]_0^1$$ because you have just done the integration as antiderivative: $F(x)=2x^{1/2} + 2x^{3/2} + \frac{10}{7}x^{7/2}$ and you have only to evaluate the primitive at the two limits of integration, so that your definite integrale is done by $F(1)-F(0)$.

• How can you say the work is correct when the final expression is blatantly wrong? I don't think it's good pedagogy to give the impression that it is just a matter of writing it differently. Precision is crucial in mathematics. – user21820 Mar 25 '17 at 9:02

We put $t=\sqrt{x}$, so $dt=\frac{1}{2\sqrt{x}}dx$. $$\int_0^12(1+3t^2+5t^6)$$ $$=2\left[t+t^3+\frac{5t^7}{7}\right]^1_0$$ $$=2\left[\sqrt{x}+\sqrt{x}^3+\frac{5\sqrt{x}^7}{7}\right]^1_0$$ $$=\frac{38}{7}$$