## Part - 1 Lecture - 6 Chapter 7 Integrals

“Don’t let yesterday take up too much of today.” – Will Rogers

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LIMIT OF SUM
$$\int_{a}^{b}{f(x)dx=\lim_{h\to0}h[f(a)+f(a+h)+f(a+2h)}+f(a+3h)+…+f(a+(n-1)h)]$$
Where, $$h=\frac{b-a}{n}$$

$$1+2+3+…+(n-1)=\frac{n(n-1)}{2}$$
$$1^2+2^2+3^2+…+(n-1)^2=\frac{n(n-1)(2n-1)}{6}$$
$$1^3+2^3+3^3+…+(n-1)^3=\left(\frac{n(n-1)}{2}\right)^2$$
$$a+ar+ar^2+…+ar^{n-1}=\frac{a(r^n-1)}{r-1}$$

$$\lim_{\theta\to0}\frac{e^\theta-1}{\theta}=1$$
$$\lim_{\theta\to0}\frac{a^\theta-1}{\theta}=log{a}$$

1. $$\int_{0}^{2}{(x^2+3)dx}=\frac{26}{3}$$
2. $$\int_{1}^{3}{(x^2+x)dx=\frac{38}{3}}$$
3. $$\int_{0}^{2}{(x^2+2x+1)dx=\frac{26}{3}}$$
4. $$\int_{0}^{4}{(x+e^{2x})dx=\frac{15+e^8}{2}}$$
5. $$\int_{1}^{3}{(2x^2+5x)dx=\frac{112}{3}}$$

6. $$\int_{1}^{3}{(2x^2+x+9)dx}=\frac{118}{3}$$
7. $$\int_{2}^{3}{e^{-4x}dx}=\frac{-1}{4}(e^{-12}-e^{-8})$$
8. $$\int_{0}^{1}e^{2-3x}dx=\frac{1}{3}\left(e^2-\frac{1}{e}\right)$$
9. $$\int_{2}^{4}2^xdx=\frac{12}{log{2}}$$
10. $$\int_{1}^{3}{(e^{2-3x}+x^2+1)dx}=\frac{1}{3}\left(32-\frac{1}{e^7}+\frac{1}{e}\right)$$