Let's solve each of the given integrals step by step.
### Q2. Evaluate the following integrals:
#### 9) [tex]\(\int_0^1 \frac{x}{\sqrt{x^2+1}} \, dx\)[/tex]
To evaluate this integral, we can use the substitution [tex]\(u = x^2 + 1 \)[/tex] which simplifies the integral. Since [tex]\(u\)[/tex] changes from 1 to 2 as [tex]\(x\)[/tex] changes from 0 to 1, we get:
[tex]\[
\int_0^1 \frac{x}{\sqrt{x^2+1}} \, dx = -1 + \sqrt{2}
\][/tex]
#### 6.) [tex]\(\int_0^1 (x + 3)^3 \, dx\)[/tex]
We can expand [tex]\((x + 3)^3\)[/tex] and then integrate term by term. The integral becomes:
[tex]\[
\int_0^1 (x + 3)^3 \, dx = \frac{175}{4}
\][/tex]
### Q3. Use the substitution method to evaluate the following integrals:
#### a) [tex]\(\int \left( \frac{\sqrt{x+2}}{\sqrt{x}} \right)^2 \, dx\)[/tex]
Simplifying the integrand:
[tex]\[
\int \left( \frac{\sqrt{x+2}}{\sqrt{x}} \right)^2 \, dx = \int \frac{x+2}{x} \, dx = \int (1 + \frac{2}{x}) \, dx = \int 1 \, dx + \int \frac{2}{x} \, dx
\][/tex]
This results in:
[tex]\[
\int \left( \frac{\sqrt{x+2}}{\sqrt{x}} \right)^2 \, dx = x + 2 \ln(x)
\][/tex]
#### b) [tex]\(\int \sin(x)(\cos(x) + 3) \, dx\)[/tex]
We can use integration by parts or simplify directly:
[tex]\[
\int \sin(x)(\cos(x) + 3) \, dx = \int (\sin(x)\cos(x) + 3\sin(x)) \, dx
\][/tex]
After integrating:
[tex]\[
\int \sin(x)(\cos(x) + 3) \, dx = \frac{\sin^2(x)}{2} - 3\cos(x)
\][/tex]
### Q4. Evaluate
#### a) [tex]\(\int \frac{1}{4+x^2} \, dx\)[/tex]
This integral is recognized as a standard form for the arctangent function:
[tex]\[
\int \frac{1}{4+x^2} \, dx = \frac{1}{2} \arctan\left(\frac{x}{2}\right)
\][/tex]
#### 5) [tex]\(\int_0^2 \frac{(x-2)}{(x+3)(x-4)} \, dx\)[/tex]
This integral can be evaluated using partial fractions, breaking it down into simpler terms. After performing partial fraction decomposition and integrating each term, we get:
[tex]\[
\int_0^2 \frac{(x-2)}{(x+3)(x-4)} \, dx = -\frac{5 \ln(3)}{7} - \frac{2 \ln(4)}{7} + \frac{2 \ln(2)}{7} + \frac{5 \ln(5)}{7}
\][/tex]
These results form the complete detailed step-by-step solutions for the given integrals.
