CLP-2 Integral Calculus

2

Suppose you want to evaluate an integral using integration by parts. You choose part of your integrand to be \(u\text{,}\) and part to be \(\dee{v}\text{.}\) The part chosen as \(u\) will be: (differentiated, antidifferentiated). The part chosen as \(\dee{v}\) will be: (differentiated, antidifferentiated).

3

Let \(f(x)\) and \(g(x)\) be differentiable functions. Using the quotient rule for differentiation, give an equivalent expression to \(\displaystyle\int \frac{f'(x)}{g(x)}\dee{x}\text{.}\)

5

Suppose you want to evaluate \(\displaystyle\int f(x)\dee{x}\) using integration by parts. Explain why \(\dee{v} = f(x)\dee{x}\text{,}\) \(u=1\) is generally a bad choice.

Note: compare this to Example 1.7.8, where we chose \(u=f(x)\text{,}\) \(\dee{v}=1\dee{x}\text{.}\)

7 (✳)

Evaluate \({\displaystyle\int \frac{\log x}{x^7}\,\dee{x}}\text{.}\)

9 (✳)

Evaluate \(\displaystyle\int_0^{\frac{\pi}{2}} x\cos x\,\dee{x}\text{.}\)

10

Evaluate \(\displaystyle\int x^3 e^x \dee{x}\text{.}\)

11

Evaluate \(\displaystyle\int x \log^3 x \dee{x}\text{.}\)

13

Evaluate \(\displaystyle\int (3t^2-5t+6)\log t\dee{t}\text{.}\)

14

Evaluate \(\displaystyle\int \sqrt{s}e^{\sqrt{s}}\dee{s}\text{.}\)

15

Evaluate \(\displaystyle\int \log^2 x \dee{x}\text{.}\)

16

Evaluate \(\displaystyle\int 2xe^{x^2+1}\dee{x}\text{.}\)

18 (✳)

Evaluate \(\displaystyle\int 4y\arctan(2y) \,\dee{y}\text{.}\)

20

Evaluate \(\displaystyle\int e^{x/2}\cos(2x)\dee{x}\text{.}\)

21

Evaluate \(\displaystyle\int \sin(\log x)\dee{x}\text{.}\)

22

Evaluate \(\displaystyle\int 2^{x+\log_2 x} \dee{x}\text{.}\)

24

Evaluate \(\displaystyle\int \dfrac{x e^{-x}}{(1-x)^2}\,\dee{x}\text{.}\)

25 (✳)

A reduction formula.

1. Derive the reduction formula
   \begin{equation*} \int\sin^n(x)\,\dee{x}=-\frac{\sin^{n-1}(x)\cos(x)}{n} +\frac{n-1}{n}\int\sin^{n-2}(x)\,\dee{x}. \end{equation*}
2. Calculate \(\displaystyle\int_0^{\pi/2}\sin^8(x)\,\dee{x}\text{.}\)

26 (✳)

Let \(R\) be the part of the first quadrant that lies below the curve \(y=\arctan x\) and between the lines \(x=0\) and \(x=1\text{.}\)

1. Sketch the region \(R\) and determine its area.
2. Find the volume of the solid obtained by rotating \(R\) about the \(y\)--axis.

27 (✳)

Let \(R\) be the region between the curves \(T(x) = \sqrt{x}e^{3x}\) and \(B(x) = \sqrt{x}(1+2x)\) on the interval \(0 \le x \le 3\text{.}\) (It is true that \(T(x)\ge B(x)\) for all \(0\le x\le 3\text{.}\)) Compute the volume of the solid formed by rotating \(R\) about the \(x\)-axis.

29

Evaluate \(\displaystyle\lim_{n \to \infty}\sum_{i=1}^n \frac{2}{n}\left(\frac{2}{n}i-1\right)e^{\frac{2}{n}i-1}\text{.}\)