Lecture 3 - Anti-Derivatives & Indefinite Integrals PDF

# BUA Faculty of Biotechnology ## Mathematics (F/R-003) ### Anti-derivative Dr. Hesham Khalaf [email protected] # Anti-Derivatives and Indefinite Integrals In order to calculate integrals, we now see that it's important to be able to find anti-derivatives of functions. An anti-derivative of...

# BUA Faculty of Biotechnology ## Mathematics (F/R-003) ### Anti-derivative Dr. Hesham Khalaf [email protected] # Anti-Derivatives and Indefinite Integrals In order to calculate integrals, we now see that it's important to be able to find anti-derivatives of functions. An anti-derivative of a function $f(x)$ is a function whose derivative is equal to $f(x)$. That is, if $F'(x) = f(x)$, then $F(x)$ is an anti- derivative of $f(x)$. Importantly, anti-derivatives are not unique. A given function can have many anti-derivatives. For instance, the following functions are all anti-derivatives of $x^2$: - $\frac{x^3}{3}$ - $\frac{x^3}{3} + 1$ - $\frac{x^3}{3} - 23$ - $\frac{x^3}{3} + \pi$ However, any two anti-derivatives of a given function differ by a constant. This allows us to write a general formula for any anti-derivative of $x^2$: $\frac{x^3}{3} + c$, where c is a constant. We write this anti-derivative as: $\int x^2 dx = \frac{x^3}{3} + c$ # Basic Integration Formulas | Formula | Description | |:---|:---| | $\int k dx = kx + c$ | $a$ is a constant | | $\int x^n dx = \frac{x^{n+1}}{n+1} + c$ | $n \neq -1$ | | $\int \frac{1}{x} dx = ln(x) + c$ | | | $\int (f(x))^n f'(x) dx = \frac{(f(x))^{n+1}}{n+1} + c$ | $n \neq -1$ | | $\int \frac{f'(x)}{f(x)} dx = ln(f(x)) + c$ | | | $\int \frac{f'(x)}{\sqrt{f(x)}} dx = 2\sqrt{f(x)}+ c$ | | | $\int e^x dx = e^x + c$ | | | $\int e^{f(x)} f'(x) dx = e^{f(x)} + c$ | | | $\int sin(x) dx = -cos(x) + c$ | | | $\int cos(x) dx = sin(x) + c$ | | | $\int sec^2(x) dx = tan(x) + c$ | | | $\int cosec^2(x) dx = -cot(x) + c$ | | | $\int sec(x) tan(x) dx = sec(x) + c$ | | | $\int cosec(x) cot(x) dx = -cosec(x) + c$ | | | $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} tan^{-1}(\frac{x}{a}) + c$ | | | $\int \frac{dx}{\sqrt{a^2 - x^2}} = sin^{-1}(\frac{x}{a}) + c$ | | # Solved Examples **Find the following integrals** 1. $\int \frac{dx}{\sqrt{1-2x}}$ 2. $\int tan^2(x) dx$ 3. $\int x(x^2-3)^5 dx$ 4. $\int e^x \sqrt{1+e^x} dx$ 5. $\int_0^1 \frac{x}{\sqrt{1- x^2}} dx$ 6. $\int_0^2 \frac{dx}{4 + x^2}$ 7. $\int_0^1 \frac{dx}{3-x}$ 8. $\int \sqrt{7x+1} dx$ 9. $\int \frac{1}{cos^2(x) \sqrt{tan(x)+1}} dx$ 10. $\int \frac{\sqrt{sin^{-1}(x)}}{\sqrt{1-x^2}} dx$ 11. $\int \frac{tan^{-1}(x)}{x^2 +1} dx$ 12. $\int x cos(x^2 +1) dx$ 13. $\int \frac{e^x + e^{-x}}{e^x - e^{-x}} dx$ 14. $\int \frac{ e^{tan^{-1}(x)}}{x^2 +1} dx$ 15. $\int \frac{dx}{\sqrt{9 - x^2}}$ 16. $\int \frac{1}{x Lnx} dx$ 17. $\int \frac{dx}{1-5x}$ # Solutions 1. $\int \frac{dx}{\sqrt{1-2x}} = -\frac{1}{2} \int \frac{-2 dx}{\sqrt{1-2x}} = -\frac{1}{2} 2\sqrt{1-2x} = -\sqrt{1-2x} + c$ 2. $\int tan^2(x) dx = \int (sec^2(x) - 1) dx = tan(x) - x + c$ 3. $\int x(x^2-3)^5 dx = \frac{1}{2} \int 2x(x^2-3)^5 dx = \frac{1}{2} \frac{(x^2-3)^6}{6} + c = \frac{(x^2-3)^6}{12} + c$ 4. $\int e^x \sqrt{1 + e^x} dx = \int e^x (1+e^x)^{1/2} dx$ =$(1 + e^x)^{3/2} / \frac{3}{2} + c = \frac{2}{3}(1 + e^x)^{3/2} + c$ 5. $\int_0^1 \frac{x}{\sqrt{1-x^2}} dx = -\frac{1}{2} \int_0^1 \frac{-2x}{\sqrt{1-x^2}} dx = -\frac{1}{2} 2\sqrt{1-x^2} |_0^1 = -(\sqrt{1-x^2})|_0^1 = -(0-1) = 1 $ 6. $\int_0^2 \frac{dx}{4+x^2} = \frac{1}{2} tan^{-1}(\frac{x}{2})|^2_0 = \frac{1}{2}(tan^{-1}(1) - tan^{-1}(0)) = \frac{1}{2}(\frac{\pi}{4} - 0) = \frac{\pi}{8}$ 7. $\int_0^1 \frac{dx}{3-x} = -\int_0^1 \frac{-dx}{3-x} = - Ln(3-x) |_0^1 = -(Ln(2) - Ln(3)) = Ln(3) - Ln(2) = Ln(\frac{3}{2})$ 8. $\int \sqrt{7x+1} dx = \frac{1}{7} \int (7x+1)^{1/2} dx$ $= \frac{1}{7} \frac{(7x+1)^{3/2}}{\frac{3}{2}}+ c = \frac{2}{21} (7x+1)^{3/2} + c$ 9. $\int \frac{1}{cos^2(x)\sqrt{tan(x)+1}} dx = \int \frac{sec^2(x)}{\sqrt{tan(x)+1}} dx = \int \frac{sec^2(x)}{\sqrt{tan(x)+1}} dx = 2\sqrt{tan(x)+1} + c$ 10. $\int \frac{\sqrt{sin^{-1}(x)}}{\sqrt{1-x^2}} dx = \int \frac{1}{\sqrt{1-x^2}}(sin^{-1}(x))^{1/2} dx= \frac{(sin^{-1}(x))^{3/2}}{\frac{3}{2}} + c$ 11. $\int \frac{tan^{-1}(x)}{x^2 +1} dx = \int \frac{1}{x^2+1} (tan^{-1}(x))dx = \frac{(tan^{-1}(x))^2}{2} + c$ 12. $\int x cos(x^2+1) dx = \frac{1}{2} \int 2x cos(x^2+1) dx$ $= \frac{1}{2} sin(x^2+1) + c$ 13. $\int \frac{e^x + e^{-x}}{e^x - e^{-x}} dx = Ln(e^x - e^{-x}) + c$ 14. $\int \frac{e^{tan^{-1}(x)}}{x^2 +1} dx = \int \frac{1}{x^2+1} e^{tan^{-1}(x)} dx = e^{tan^{-1}(x)} + c$ 15. $\int \frac{dx}{\sqrt{9-x^2}} = sin^{-1}(\frac{x}{3}) + c$ 16. $\int \frac{1}{x Lnx} dx = \int \frac{1}{Lnx} \frac{1}{x} dx = Ln(Ln(x)) + c$ 17. $\int \frac{dx}{1-5x} = -\frac{1}{5} \int \frac{-5 dx}{1-5x} = -\frac{1}{5} Ln(1-5x) + c$ # Finish **See you next week**