1. ## Indefinite Integral Help

Can you help show how to solve these??

1. (x^2/3) - (x^1/4) / (x^1/2) dx
2. [x(x+2)] / (x^3 + 3x^2 - 5) dx
3. (X^3)(x^4 - 4x^2 + 4)^1/2 dx
4. [(x-1)/(x^5)]^1/2 dx
5. (x^2 + 3)^1/3 (x^5) dx
6. (x^3 + 3x^2 - 9) / (x + 3)^1/2 dx

2. ## Re: Indefinite Integral Help

1. Write it as \displaystyle \displaystyle \begin{align*} x^{\frac{2}{3}} - x^{-\frac{1}{4}} \end{align*} and integrate term by term.

2.
\displaystyle \displaystyle \begin{align*} \int{\frac{x(x + 2)}{x^3 + 3x^2 - 5}\,dx} &= \int{\frac{x^2 + 2x}{x^3 + 3x^2 - 5}\,dx} \\ &= \frac{1}{3} \int{ \left( \frac{3x^2 + 6x}{x^3 + 3x^2 - 5} \right) dx } \end{align*}

Now you can make the substitution \displaystyle \displaystyle \begin{align*} u = x^3 + 3x^2 - 5 \implies du = 3x^2 + 6x \, dx \end{align*} and integrate.

3.
\displaystyle \displaystyle \begin{align*} \int{\frac{x^3}{\sqrt{x^4 - 4x^2 + 4}}\,dx} &= \int{\frac{x^3}{\sqrt{\left( x^2 - 2 \right)^2 }} \, dx} \\ &= \int{ \frac{x^3}{ x^2 - 2 }\, dx} \\ &= \frac{1}{2} \int{ \frac{2x \cdot x^2 }{x^2 - 2} \, dx } \end{align*}

Now make the substitution \displaystyle \displaystyle \begin{align*} u = x^2 - 2 \implies du = 2x \, dx \end{align*}.

3. ## Re: Indefinite Integral Help

4.

$\displaystyle \int \sqrt{\frac{x-1}{x^5}}dx=\int \frac{\sqrt{x-1}}{x^{2}\sqrt{x}}dx$
$\displaystyle =\int \frac{1}{x^2}\sqrt{\frac{x-1}{x}}dx$ now take $\displaystyle \frac{x-1}{x}=t^2 \implies \frac{dx}{x^2}=2tdt$
now the integral becomes $\displaystyle \int 2t^{2}dt = 2\frac{t^3}{3}=\frac{2}{3}(\frac{x-1}{x})^{\frac{3}{2}}$

6.

$\displaystyle \int \frac{x^{3}+3x^{2}-9}{\sqrt{x+3}}dx=\int\frac{x^{2}(x+3)-9}{\sqrt{x+3}}dx$
now we integrate separately
$\displaystyle I_{1}=\int\frac{x^{2}(x+3)}{\sqrt{x+3}}dx=\int x^{2}\sqrt{x+3}dx$ take $\displaystyle x+3=t^2 \implies dx=2tdt & x^2=(t^2-3)^{2}$
$\displaystyle I_{1}=\int (t^2-3)^{2}t^{2}dt=2\int (t^{6}-t^{4}+9t^{2})=2[\frac{t^7}{7}+\frac{6t^5}{5}+\frac{9t^3}{3}]$
substitute back the value of t to get the value in $\displaystyle x$
$\displaystyle I_{2}=\int \frac{9}{\sqrt{x+3}}$ take $\displaystyle x+3=t^2$ and $\displaystyle I_{2}$ becomes $\displaystyle 18(x+3)^{\frac{1}{2}}$
Add $\displaystyle I_{1}$ and $\displaystyle I_{2}$ to get the result

5.

Well I tried for sometime & could not do
Spoiler:
I approached it like the first one $\displaystyle \int\frac{\sqrt[3]{x^2+3}}{x^5}dx$$\displaystyle =\int\sqrt[3]{\frac{x^2+3}{x^3}}\frac{dx}{x^4}$
take $\displaystyle \frac{x^2+3}{x^3}=t^2 \implies \frac{dx}{x^4}=-\frac{3t^2}{x^2+9}$ now we can try expressing x in terms of t..which wont work i guess...

Whats the answer ?, does it even have a elementary anti derivative?

4. ## Re: Indefinite Integral Help

Originally Posted by vanillachyrae
Can you help show how to solve these??

1. (x^2/3) - (x^1/4) / (x^1/2) dx
2. [x(x+2)] / (x^3 + 3x^2 - 5) dx
3. (X^3)(x^4 - 4x^2 + 4)^1/2 dx
4. [(x-1)/(x^5)]^1/2 dx
5. (x^2 + 3)^1/3 (x^5) dx
6. (x^3 + 3x^2 - 9) / (x + 3)^1/2 dx

5.
\displaystyle \displaystyle \begin{align*} \int{x^5 \left( x^2 + 3 \right)^{\frac{1}{3}} \, dx} &= \frac{1}{2} \int{ 2x \cdot x^4 \left( x^2 + 3 \right)^{\frac{1}{3}}\,dx } \end{align*}

Now let \displaystyle \displaystyle \begin{align*} u = x^2 + 3 \implies du = 2x \, dx \end{align*} and the integral becomes

\displaystyle \displaystyle \begin{align*} \frac{1}{2}\int{ 2x \cdot x^4 \left( x^2 + 3 \right)^{\frac{1}{3}}\, dx} &= \frac{1}{2} \int{ \left( u - 3 \right)^2 \, u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \int{ \left( u^2 - 6u + 9 \right) u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \int{ u^{\frac{7}{3}} - 6u^{\frac{4}{3}} + 9u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \left( \frac{3}{10}u^{\frac{10}{3}} - \frac{18}{7}u^{\frac{7}{3}} + \frac{27}{4} u^{\frac{4}{3}} \right) + C \\ &= \frac{3}{20} \left( x^2 + 3 \right)^{\frac{10}{3}} - \frac{9}{7} \left( x^2 + 3 \right)^{\frac{7}{3}} + \frac{27}{8} \left( x^2 + 3 \right)^{\frac{4}{3}} + C \end{align*}

5. ## Re: Indefinite Integral Help

Q No 5 Attached

6. ## Re: Indefinite Integral Help

Originally Posted by ibdutt
Q No 5 Attached
Is this not exactly what I did in the post directly above?

7. ## Re: Indefinite Integral Help

Hmm..seems like I got the whole question 5 wrong...
@ vanillachyrae : please learn to write in latex(its not at all hard)
@ Prove it: You were great and quick as usual

8. ## Re: Indefinite Integral Help

Originally Posted by Prove It
5.
\displaystyle \displaystyle \begin{align*} \int{x^5 \left( x^2 + 3 \right)^{\frac{1}{3}} \, dx} &= \frac{1}{2} \int{ 2x \cdot x^4 \left( x^2 + 3 \right)^{\frac{1}{3}}\,dx } \end{align*}

Now let \displaystyle \displaystyle \begin{align*} u = x^2 + 3 \implies du = 2x \, dx \end{align*} and the integral becomes

\displaystyle \displaystyle \begin{align*} \frac{1}{2}\int{ 2x \cdot x^4 \left( x^2 + 3 \right)^{\frac{1}{3}}\, dx} &= \frac{1}{2} \int{ \left( u - 3 \right)^2 \, u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \int{ \left( u^2 - 6u + 9 \right) u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \int{ u^{\frac{7}{3}} - 6u^{\frac{4}{3}} + 9u^{\frac{1}{3}}\, du } \\ &= \frac{1}{2} \left( \frac{3}{10}u^{\frac{10}{3}} - \frac{18}{7}u^{\frac{7}{3}} + \frac{27}{4} u^{\frac{4}{3}} \right) + C \\ &= \frac{3}{20} \left( x^2 + 3 \right)^{\frac{10}{3}} - \frac{9}{7} \left( x^2 + 3 \right)^{\frac{7}{3}} + \frac{27}{8} \left( x^2 + 3 \right)^{\frac{4}{3}} + C \end{align*}
Also

\displaystyle 5.\displaystyle \begin{align*} \int{x^5 \left( x^2 + 3 \right)^{\frac{1}{3}} \, dx} \end{align*}
now let
\displaystyle \displaystyle \begin{align*} t = x^2 \implies dt = 2x \, dx \end{align*} and the integral becomes

$\displaystyle \frac{1}{2}\int{ t^2(t+3)^{\frac{1}{3}}dt}$

by parts
let $\displaystyle u=t^2$ and $\displaystyle dv=(t+3)^{\frac{1}{3}}dt$
and once again with integration by parts