# Understanding Definite and Indefinite Integrals
## Definite Integrals
A definite integral is a fundamental concept in calculus that provides a way to calculate the accumulation of quantities and represents the exact area under a curve within a specific interval. Specifically, if you have a function f(x) that is continuous on the interval [a, b], then the definite integral of f(x) from a to b is given by:
`\[ \int_{a}^{b} f(x) \, dx \]`
This integral can be interpreted as the net area between the function f(x) and the x-axis, from x = a to x = b.
The Fundamental Theorem of Calculus connects the concept of differentiation with that of integration and states that if F(x) is an antiderivative of f(x), then:
`\[ \int_{a}^{b} f(x) \, dx = F(b) – F(a) \]`
This means that the value of the definite integral is the difference between the values of an antiderivative at the endpoints of the interval.
## Indefinite Integrals
In contrast to definite integrals, indefinite integrals represent a family of functions rather than a numerical value. The indefinite integral, often interpreted as the antiderivative, is given by the general formula:
`\[ \int f(x) \, dx = F(x) + C \]`
where F(x) is the antiderivative of f(x), and C represents a constant of integration. Each member of the family of functions described by the indefinite integral differs only by a constant term.
## Solving Problems: Definite and Indefinite Integrals
Here are 20 problems with their respective solutions involving definite and indefinite integrals.
### Problems and Solutions – Indefinite Integrals
1. **Problem:** Integrate `\(\int (3x^2 – 2x + 4) \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int (3x^2 – 2x + 4) \, dx &= \int 3x^2 \, dx – \int 2x \, dx + \int 4 \, dx \\
&= x^3 – x^2 + 4x + C
\end{align*}
\]
“`
2. **Problem:** Integrate `\(\int \frac{1}{x} \, dx\)`.
**Solution:**
“`
\[
\int \frac{1}{x} \, dx = \ln|x| + C
\]
“`
3. **Problem:** Integrate `\(\int e^x \, dx\)`.
**Solution:**
“`
\[
\int e^x \, dx = e^x + C
\]
“`
4. **Problem:** Integrate `\(\int \cos(x) \, dx\)`.
**Solution:**
“`
\[
\int \cos(x) \, dx = \sin(x) + C
\]
“`
5. **Problem:** Integrate `\(\int x(x^2 + 1) \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int x(x^2 + 1) \, dx &= \int (x^3 + x) \, dx \\
&= \frac{1}{4}x^4 + \frac{1}{2}x^2 + C
\end{align*}
\]
“`
6. **Problem:** Integrate `\(\int \sec^2(x) \, dx\)`.
**Solution:**
“`
\[
\int \sec^2(x) \, dx = \tan(x) + C
\]
“`
7. **Problem:** Integrate `\(\int \sqrt{x} \, dx\)`.
**Solution:**
“`
\[
\int \sqrt{x} \, dx = \frac{2}{3}x^{\frac{3}{2}} + C
\]
“`
8. **Problem:** Integrate `\(\int \sin^2(x) \, dx\)` using the identity `\(sin^2(x) = \frac{1}{2}(1 – \cos(2x))\)`.
**Solution:**
“`
\[
\begin{align*}
\int \sin^2(x) \, dx &= \int \frac{1}{2}(1 – \cos(2x)) \, dx \\
&= \frac{1}{2}x – \frac{1}{4}\sin(2x) + C
\end{align*}
\]
“`
9. **Problem:** Integrate `\(\int \frac{1}{\sqrt{1-x^2}} \, dx\)`.
**Solution:**
“`
\[
\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C
\]
“`
10. **Problem:** Integrate `\(\int \ln(x) \, dx\)` using integration by parts.
**Solution:**
“`
\[
\begin{align*}
\int \ln(x) \, dx &= x \ln(x) – \int x \frac{1}{x} \, dx \\
&= x \ln(x) – x + C
\end{align*}
\]
“`
### Problems and Solutions – Definite Integrals
11. **Problem:** Evaluate `\(\int_{0}^{1} (x^3 – x^2) \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{1} (x^3 – x^2) \, dx &= \left[ \frac{x^4}{4} – \frac{x^3}{3} \right]_0^1 \\
&= \frac{1}{4} – \frac{1}{3} \\
&= -\frac{1}{12}
\end{align*}
\]
“`
12. **Problem:** Evaluate `\(\int_{-1}^{1} e^x \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{-1}^{1} e^x \, dx &= [e^x]_{-1}^{1} \\
&= e – \frac{1}{e} \\
&= e – e^{-1}
\end{align*}
\]
“`
13. **Problem:** Evaluate `\(\int_{0}^{\pi/2} \cos(x) \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{\pi/2} \cos(x) \, dx &= [\sin(x)]_{0}^{\pi/2} \\
&= 1 – 0 \\
&= 1
\end{align*}
\]
“`
14. **Problem:** Evaluate `\(\int_{1}^{2} \frac{1}{x^2} \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{1}^{2} \frac{1}{x^2} \, dx &= \left[ -\frac{1}{x} \right]_1^2 \\
&= -\frac{1}{2} + 1 \\
&= \frac{1}{2}
\end{align*}
\]
“`
15. **Problem:** Evaluate `\(\int_{0}^{\pi} x \sin(x) \, dx\)` using integration by parts.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{\pi} x \sin(x) \, dx &= -x \cos(x) \bigg|_0^{\pi} + \int_{0}^{\pi} \cos(x) \, dx \\
&= -\pi(-1) + [ \sin(x) ]_0^\pi \\
&= \pi
\end{align*}
\]
“`
16. **Problem:** Evaluate `\(\int_{0}^{1} 3x^2 \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{1} 3x^2 \, dx &= [x^3]_0^1 \\
&= 1^3 – 0^3 \\
&= 1
\end{align*}
\]
“`
17. **Problem:** Evaluate `\(\int_{1/e}^{e} \frac{1}{x} \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{1/e}^{e} \frac{1}{x} \, dx &= [\ln|x|]_{1/e}^{e} \\
&= \ln(e) – \ln(1/e) \\
&= 1 + 1 \\
&= 2
\end{align*}
\]
“`
18. **Problem:** Evaluate `\(\int_{0}^{2\pi} \sin^2(x) \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{2\pi} \sin^2(x) \, dx &= \frac{\pi}{2}
\end{align*}
\]
“`
19. **Problem:** Evaluate `\(\int_{0}^{4} \sqrt{x} \, dx\)`.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{4} \sqrt{x} \, dx &= \frac{2}{3}[x^{\frac{3}{2}}]_0^4 \\
&= \frac{2}{3}(8 – 0) \\
&= \frac{16}{3}
\end{align*}
\]
“`
20. **Problem:** Evaluate `\(\int_{0}^{1} \ln(x + 1) \, dx\)` using integration by parts.
**Solution:**
“`
\[
\begin{align*}
\int_{0}^{1} \ln(x + 1) \, dx &= (x + 1) \ln(x + 1) \bigg|_0^{1} – \int_{0}^{1} \frac{x + 1}{x + 1} \, dx \\
&= (2 \ln(2) – 0) – [x]_0^1 \\
&= 2 \ln(2) – 1
\end{align*}
\]
“`
The above problems and solutions cover a variety of integral types and provide a solid practice ground for anyone looking to master the calculation of both definite and indefinite integrals.
