Methods of Integration Solved Exercises

This text presents a collection of 35 solved exercises from the "Higher Education EXTRA-MATH SERIES" authored by Hussein Raad.
The exercises focus on fundamental and advanced methods of integration, including linear substitution, substitution for transcendental functions, and integration by parts.

Exercise 1: $\int \sqrt{(2x+1)^3} \,dx$ - Features a linear radical function where substitution $u = 2x + 1$ is applicable.
Exercise 2: $\int \frac{4}{(3-2x)^5} \,dx$ - Involves a reciprocal power of a linear expression; utilizes the rule $\int (ax+b)^n \,dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C$ .
Exercise 3: $\int 2x \sqrt{x^2-1} \,dx$ - A basic substitution case where $u = x^2 - 1$ and $du = 2x \,dx$ .
Exercise 5: $\int 3x^2 \sqrt{x^3+4} \,dx$ - Demonstrates the substitution $u = x^3 + 4$ , where the outer factor $3x^2$ matches the derivative of the inner function.
Exercise 14: $\int \frac{3}{x^2(\frac{1}{x}+1)^4} \,dx$ - A multi-step algebraic substitution where $u = \frac{1}{x} + 1$ leads to $du = -\frac{1}{x^2} \,dx$ .
Exercise 23: $\int \frac{dx}{\sqrt{x}+1}$ - Rationalization or substitution with $u = \sqrt{x}$ is required.
Exercise 24: $\int \frac{x}{1-x} \,dx$ - Solved via algebraic manipulation (adding and subtracting 1 in the numerator) or substitution $u = 1-x$ .
Exercise 26: $\int \frac{x^3}{4+x^2} \,dx$ - Requires algebraic long division or the substitution $u = 4+x^2$ .

Exercise 4: $\int (x+2) \sin(x^2 + 4x - 6) \,dx$ - The derivative of the inner quadratic function $x^2 + 4x - 6$ is $2x + 4$ , which is a multiple of the outer factor $(x+2)$ .
Exercise 6: $\int \frac{\cot(\ln(x))}{x} \,dx$ - Substitution $u = \ln(x)$ . Transforms the integral into the standard form $\int \cot(u) \,du$ .
Exercise 11: $\int \frac{\cos(\sqrt{x}+3)}{\sqrt{x}} \,dx$ - Uses $u = \sqrt{x}+3$ ; note that $du = \frac{1}{2\sqrt{x}} \,dx$ .
Exercise 12: $\int \frac{\tan(\sqrt{x})}{\sqrt{x}} \,dx$ - Similar to Exercise 11, focusing on the tangent function.
Exercise 13: $\int 2^{-x} \tanh(2^{1-x}) \,dx$ - A complex substitution involving exponential bases and hyperbolic functions. Often requires $u = 2^{1-x}$ .
Exercise 15: $\int e^x \sin(e^x) \,dx$ - Direct substitution of the exponential inner function: $u = e^x$ .
Exercise 18: $\int 2 \sec^2(4x) \,dx$ - Direct application of the integral of $\sec^2(ax)$ .
Exercise 34: $\int x \tan(x^2) \sec(x^2) \,dx$ - Uses $u = x^2$ , resulting in the standard integral of $\sec(u)\tan(u)$ , which is $\sec(u) + C$ .
Exercise 35: $\int \frac{dx}{\sin(x) \cos(x)}$ - Can be solved using trigonometric identities (e.g., multiplying by $\frac{\sec^2(x)}{\sec^2(x)}$ or using the double angle identity $\sin(2x) = 2 \sin(x) \cos(x)$ ).

Exercise 7: $\int x^2 e^{x^3} \,dx$ - Chain rule inverse with $u = x^3$ .
Exercise 8: $\int e^{e^x+x} \,dx$ - Relies on the property $e^{a+b} = e^a e^b$ . The expression simplifies to $\int e^{e^x} e^x \,dx$ , allowing for $u = e^x$ .
Exercise 9: $\int \frac{e^x}{1+e^x} \,dx$ - Results in the natural log form $\ln|1+e^x| + C$ .
Exercise 10: $\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \,dx$ - Substitution $u = \sqrt{x}$ .
Exercise 21: $\int \frac{e^x-2x}{e^x-x^2} \,dx$ - A logarithmic form integral where the numerator is the derivative of the denominator: $\frac{d}{dx}(e^x - x^2) = e^x - 2x$ .
Exercise 27: $\int \frac{e^x}{1+e^{2x}} \,dx$ - Leads to the arctangent form $\arctan(e^x) + C$ by substituting $u = e^x$ .
Exercise 28: $\int \ln\left(\frac{1}{x}\right) \,dx$ - Simplified using log properties as $\int -\ln(x) \,dx$ , then integrated by parts.
Exercise 29: $\int \frac{3}{x \ln^2(x)} \,dx$ - Substitution $u = \ln(x)$ transforms this into a power rule integral $\int 3 u^{-2} \,du$ .
Exercise 33: $\int \frac{\sqrt{\ln(x)}}{x} \,dx$ - Substitution $u = \ln(x)$ .

Exercise 16: $\int (3x+2) e^x \,dx$ - Let $u = 3x+2$ and $dv = e^x \,dx$ .
Exercise 17: $\int x e^{-2x} \,dx$ - Let $u = x$ and $dv = e^{-2x} \,dx$ .
Exercise 19: $\int (2x-1) \cos(x) \,dx$ - Let $u = 2x-1$ and $dv = \cos(x) \,dx$ .
Exercise 20: $\int (1-x) \sin(x) \,dx$ - Let $u = 1-x$ and $dv = \sin(x) \,dx$ .
Exercise 30: $\int \ln(x) \,dx$ - Standard entry-level integration by parts where $u = \ln(x)$ and $dv = dx$ .
Exercise 31: $\int x \ln(x) \,dx$ - Part of the logarithmic series; let $u = \ln(x)$ and $dv = x \,dx$ .
Exercise 32: $\int \frac{\ln(x)}{x^2} \,dx$ - Let $u = \ln(x)$ and $dv = x^{-2} \,dx$ .

Exercise 22: $\int \frac{x}{1+x^4} \,dx$ - Substitution $u = x^2$ converts this to the form $\int \frac{1}{1+u^2} \,du$ , leading to $\frac{1}{2}\arctan(x^2) + C$ .
Exercise 25: $\int \frac{\arcsin(x)}{\sqrt{1-x^2}} \,dx$ - Substitution $u = \arcsin(x)$ simplifies the expression significantly as $du = \frac{1}{\sqrt{1-x^2}} \,dx$ .