We need to evaluate two indefinite integrals. a) $\int \frac{x^2}{2} dx$ c) $\int \frac{1}{(x + \frac{2}{3})(x - \frac{4}{5})} dx$

1. Problem Description

We need to evaluate two indefinite integrals.

a) $\int \frac{x^2}{2} dx$

c) $\int \frac{1}{(x + \frac{2}{3})(x - \frac{4}{5})} dx$

2. Solution Steps

a) $\int \frac{x^2}{2} dx$

We can rewrite the integral as:

$\frac{1}{2} \int x^2 dx$

Using the power rule for integration:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

where $n$ is a constant and $C$ is the constant of integration.

In this case, $n = 2$.

So, $\frac{1}{2} \int x^2 dx = \frac{1}{2} \cdot \frac{x^{2+1}}{2+1} + C = \frac{1}{2} \cdot \frac{x^3}{3} + C = \frac{x^3}{6} + C$

c) $\int \frac{1}{(x + \frac{2}{3})(x - \frac{4}{5})} dx$

We can use partial fraction decomposition to simplify the integrand.

Let $\frac{1}{(x + \frac{2}{3})(x - \frac{4}{5})} = \frac{A}{x + \frac{2}{3}} + \frac{B}{x - \frac{4}{5}}$

Multiplying both sides by $(x + \frac{2}{3})(x - \frac{4}{5})$ gives:

$1 = A(x - \frac{4}{5}) + B(x + \frac{2}{3})$

To solve for $A$ and $B$, we can use the following method:

Set $x = \frac{4}{5}$:

$1 = A(\frac{4}{5} - \frac{4}{5}) + B(\frac{4}{5} + \frac{2}{3})$

$1 = 0 + B(\frac{12 + 10}{15})$

$1 = B(\frac{22}{15})$

$B = \frac{15}{22}$

Set $x = -\frac{2}{3}$:

$1 = A(-\frac{2}{3} - \frac{4}{5}) + B(-\frac{2}{3} + \frac{2}{3})$

$1 = A(\frac{-10 - 12}{15}) + 0$

$1 = A(\frac{-22}{15})$

$A = -\frac{15}{22}$

So, the integral becomes:

$\int \frac{1}{(x + \frac{2}{3})(x - \frac{4}{5})} dx = \int \left( \frac{-\frac{15}{22}}{x + \frac{2}{3}} + \frac{\frac{15}{22}}{x - \frac{4}{5}} \right) dx$

$= -\frac{15}{22} \int \frac{1}{x + \frac{2}{3}} dx + \frac{15}{22} \int \frac{1}{x - \frac{4}{5}} dx$

Using the fact that $\int \frac{1}{x+a} dx = \ln|x+a| + C$, we get:

$= -\frac{15}{22} \ln|x + \frac{2}{3}| + \frac{15}{22} \ln|x - \frac{4}{5}| + C$

$= \frac{15}{22} \left( \ln|x - \frac{4}{5}| - \ln|x + \frac{2}{3}| \right) + C$

$= \frac{15}{22} \ln\left| \frac{x - \frac{4}{5}}{x + \frac{2}{3}} \right| + C$

3. Final Answer

a) $\frac{x^3}{6} + C$

c) $\frac{15}{22} \ln\left| \frac{x - \frac{4}{5}}{x + \frac{2}{3}} \right| + C$