Chapter 7 - Principles Of Integral Evaluation - 7.2 Integration By Parts - Exercises Set 7.2 - Page 499: 65

$$\eqalign{ & \left( {\text{a}} \right)\frac{{{{\tan }^3}x}}{3} - \tan x + x + C \cr & \left( {\text{b}} \right)\frac{{{{\sec }^2}x\tan x}}{3} + \frac{2}{3}\tan x + C \cr & \left( {\text{c}} \right){x^2}{e^x} - 2{x^2}{e^x} + 4x{e^x} - 4{e^x} + C \cr} $$

$$\eqalign{ & \left( {\text{a}} \right)\int {{{\tan }^4}x} dx \cr & {\text{Integrate using the reduction formula}} \cr & \int {{{\tan }^n}x} dx = \frac{{{{\tan }^{n - 1}}x}}{{n - 1}} - \int {{{\tan }^{n - 2}}x} dx \cr & \int {{{\tan }^4}x} dx = \frac{{{{\tan }^3}x}}{3} - \int {{{\tan }^2}x} dx \cr & {\text{Use pythagorean identities}} \cr & \int {{{\tan }^4}x} dx = \frac{{{{\tan }^3}x}}{3} - \int {\left( {{{\sec }^2}x - 1} \right)} dx \cr & \int {{{\tan }^4}x} dx = \frac{{{{\tan }^3}x}}{3} - \int {{{\sec }^2}x} dx + \int {dx} \cr & \int {{{\tan }^4}x} dx = \frac{{{{\tan }^3}x}}{3} - \tan x + x + C \cr & \cr & \left( {\text{b}} \right)\int {{{\sec }^4}x} dx \cr & {\text{Integrate using the reduction formula}} \cr & \int {{{\sec }^n}x} dx = \frac{{{{\sec }^{n - 2}}x\tan x}}{{n - 1}} + \frac{{n - 2}}{{n - 1}}\int {{{\sec }^{n - 2}}x} dx \cr & \int {{{\sec }^4}x} dx = \frac{{{{\sec }^2}x\tan x}}{3} + \frac{2}{3}\int {{{\sec }^{4 - 2}}x} dx \cr & {\text{Simplifying}} \cr & \int {{{\sec }^4}x} dx = \frac{{{{\sec }^2}x\tan x}}{3} + \frac{2}{3}\int {{{\sec }^2}x} dx \cr & \int {{{\sec }^4}x} dx = \frac{{{{\sec }^2}x\tan x}}{3} + \frac{2}{3}\tan x + C \cr & \cr & \left( {\text{c}} \right)\int {{x^3}{e^x}} dx \cr & {\text{Integrate using the reduction formula}} \cr & \int {{x^n}{e^x}} dx = {x^n}{e^x} - n\int {{x^{n - 1}}{e^x}} dx \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2\int {{x^{3 - 1}}{e^x}} dx \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2\int {{x^2}{e^x}} dx \cr & {\text{Let }}n = 2 \cr & {\text{Simplifying}} \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2\left( {{x^2}{e^x} - 2\int {{x^{2 - 1}}{e^x}} dx} \right) \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2{x^2}{e^x} + 4\int {x{e^x}} dx \cr & {\text{Let }}n = 1 \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2{x^2}{e^x} + 4\left( {x{e^x} - \int {{e^x}} dx} \right) \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2{x^2}{e^x} + 4x{e^x} - 4\int {{e^x}} dx \cr & \int {{x^2}{e^x}} dx = {x^2}{e^x} - 2{x^2}{e^x} + 4x{e^x} - 4{e^x} + C \cr} $$