Calculus Solutions EX#5.4

  Question 1. Evaluate.   (a) $$ \begin{aligned} \sum_{k=1}^{3} k^{3}=1^{3}+2^{3}+3^{3}=1+8+27=36 \end{aligned} $$ (b) $$ \begin{aligned} \sum_{j=2}^{6}(3 j-1)= & (3(2)-1)+(3(3)-1)+(3(4)-1)+(3(5)-1) \\ \\ & +(3(6)-1) \\ \\. = & (6-1)+(9-1)+(12-1)+(15-1)+(18-1) \\ \\ = & 5+8+11+14+17=55 \\ \\ \end{aligned} $$ (c) $$ \begin{aligned} \sum_{i=-4}^{1}(i^2 -i)= ((-4)^{2}+4)+((-3)^{2}-(-3))+((-2)^{2}-(-2))+\\ \\ ((-1){^2}-(-1))+((0)^{2}-0)+((1)^{2}-1) \\ \\ = (16+4)+(9+3)+(4+2)+(1+1)+0+(1-1) \\ \\ = 20+12+6+2=40 \end{aligned} $$ … Read more

Calculus Solution Ex#5.3

    EXERCISE $5 \cdot 3$ Evaluate the integrals using appropriate Substitutions $15-$ $$ \begin{aligned} \int(4 x-3)^{9} d x \\ \\ \text { Let } u=4 x-3  \quad \quad \quad (1) \\ \\ d u=4 d x \\ \\ \frac{1}{4} d u=d x \end{aligned} $$ then the above integral becomes $$ \begin{aligned} \frac{1}{4} \int u^{9} … Read more

Calculus Solution Ex #5.2

In this section, we shall develop the concept of antiderivatives. It contains solutions to integration problems. (1) — In each part, confirm that the formula is correct…. (a) $$ \begin{aligned} & \frac{d}{d x}\left[\sqrt{1+x^{2}}\right] \\ \\ & =\frac{d}{d x}\left[1+x^{2}\right]^{1 / 2} \\ \\ & =\frac{1}{2}\left[1+x^{2}\right]^{-1 / 2}(2 x) \\ \\ & =\frac{x}{\left(1+x^{2}\right)^{1 / 2}}=\frac{x}{\sqrt{1+x^{2}}} \end{aligned} $$ corresponding … Read more