1. ## Conceptual Integrals

$\displaystyle \int h(x)f^1(x) * dx = \int$

and

$\displaystyle \int \frac{2}{3x^2 + 4x + 5} * dx = \int$

2. Originally Posted by Selim
$\displaystyle \int h(x)f^1(x) * dx = \int$

and

$\displaystyle \int \frac{2}{3x^2 + 4x + 5} * dx = \int$

your second question is very easy. express the denominator in the form x^2+-a^2 where a is a constant. then integrate.

3. Originally Posted by Selim
$\displaystyle \int h(x)f^1(x) * dx = \int$

and

$\displaystyle \int \frac{2}{3x^2 + 4x + 5} * dx = \int$
The bottom does not factorise, so you can't use partial fractions.

That means you'll have to complete the square and use a trigonometric substitution.

$\displaystyle \int{\frac{2}{3x^2 + 4x + 5}\,dx} = \frac{2}{3}\int{\frac{1}{x^2 + \frac{4}{3}x + \frac{5}{3}}\,dx}$

$\displaystyle = \frac{2}{3}\int{\frac{1}{x^2 + \frac{4}{3}x + \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2 + \frac{5}{3}}\,dx}$

$\displaystyle = \frac{2}{3}\int{\frac{1}{\left(x + \frac{2}{3}\right)^2 + \frac{11}{9}}\,dx}$

Now make the substitution $\displaystyle x + \frac{2}{3} = \sqrt{\frac{11}{9}}\tan{\theta} = \frac{\sqrt{11}}{3}\tan{\theta}$.

Note that $\displaystyle dx = \frac{\sqrt{11}}{3}\sec^2{\theta}\,d\theta$.

Also note that $\displaystyle \theta = \arctan{\left[\frac{3\sqrt{11}}{11}\left(x + \frac{2}{3}\right)\right]}$.

Then the integral becomes

$\displaystyle = \frac{2}{3}\int{\frac{1}{\left(\frac{\sqrt{11}}{3} \tan{\theta}\right)^2 + \frac{11}{9}}\,\frac{\sqrt{11}}{3}\sec^2{\theta}\, d\theta}$

$\displaystyle = \frac{2}{3}\int{\frac{\frac{\sqrt{11}}{3}\sec^2{\t heta}}{\frac{11}{9}\tan^2{\theta} + \frac{11}{9}}\,d\theta}$

$\displaystyle = \frac{2}{3}\int{\frac{\frac{\sqrt{11}}{3}\sec^2{\t heta}}{\frac{11}{9}(\tan^2{\theta} + 1)}\,d\theta}$

$\displaystyle = \frac{2\sqrt{11}}{11}\int{\frac{\sec^2{\theta}}{\s ec^2{\theta}}\,d\theta}$

$\displaystyle = \frac{2\sqrt{11}}{11}\int{1\,d\theta}$

$\displaystyle = \frac{2\sqrt{11}}{11}\theta + C$

$\displaystyle = \frac{2\sqrt{11}}{11}\arctan{\left[\frac{3\sqrt{11}}{11}\left(x + \frac{2}{3}\right)\right]} + C$.

4. Thanks for that messy one, sir, and the other one is pretty easy,

$\displaystyle \int h(x) * f^1(x) * dx = h(x) * f(x) - \int f(x) * h^1(x)$

I was simply supposed to integrate it by parts.