Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 301: 65

a)$-1$ b)$\frac{1}{12}$ c)$-4$

a) $T(3x^2-2)=\int_{0}^{1} 3x^2-2 dx=x^3-2x\Big|_0^1=(1^3-2(1))-(0^3-2(0))=-1-0=-1$ b) $T(x^3-x^5)=\int_{0}^{1} x^3-x^5dx=\frac{x^4}{4}-\frac{x^6}{6}\Big|_0^1=(\frac{1^4}{4}-\frac{1^6}{6})-(\frac{0^4}{4}-\frac{0^6}{6})=\frac{1}{12}-0=\frac{1}{12}$ c) $T(4x-6)=\int_{0}^{1} 4x-6 dx=2x^2-6x\Big|_0^1=(2(1)^2-6(1))-(2(0)^2-6(0))=-4-0=-4$