# Question 13.1: Given  dy / dx = cos x − x, find y....

Given  $\frac{\mathrm{d} y}{\mathrm{~d} x}=\cos x-x, \text { find } y .$

Step-by-Step
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We need to find a function which, when differentiated, yields $\cos x − x$. Differentiating sin x yields $\cos x$, while differentiating −x²/2 yields −x. Hence,

$y=\int(\cos x-x) \mathrm{d} x=\sin x-\frac{x^2}{2}+c$

where c is the constant of integration. Usually brackets are not used and the integral is written simply as  $\int \cos x-x d x$.

Question: 13.11

## Find the area contained by y = sin x from x = 0 to x = 3π/2. ...

Figure 13.13 illustrates the required area. From t...
Question: 13.10

## (a) Sketch y = sin x for x = −π to x = π. (b) Calculate ∫^π -π sin x dx and comment on your findings. (c) Calculate the area enclosed by y = sin x and the x axis between x = −π and x = π. ...

(a) A graph of y = sin x between x = −π and x = π ...
Question: 13.9

## Find the area bounded by y = x³ and the x axis from x = −3 to x = −2. ...

Figure 13.11 illustrates the required area. [latex...
Question: 13.8

## Find the area under z(t ) = e^2t from t = 1 to t = 3. ...

\begin{aligned}\text { Area } & =\int_1...
Question: 13.7

## Evaluate (a) ∫²1 x² + 1 dx  (b) ∫¹2 x² + 1 dx (c) ∫^π 0 sin x dx ...

(a) Let I stand for  \int_1^2 x^2+1 d x[/la...
Question: 13.6

## Find (a) ∫ sin 2t cos t dt (b) ∫ sin mt sin nt dt, where m and n are constants with m ≠ n ...

(a) Using the identities in Table 3.1 we find [lat...
Question: 13.5

## Evaluate (a) ∫ cos² t dt (b) ∫ sin² t dt ...

Powers of trigonometric functions, for example  [l...
Question: 13.4

## Use Table 13.1 and the properties of a linear operator to integrate the following expressions: (a) x² + 9 (b) 3t^4 −√t (c) 1 / x (d) (t + 2)² (e) 1 / z + z (f) 4e^2z (g) 3 sin 4t (h) 4 cos(9x + 2) (i) 3e^2z (j) sin x + cos x / 2 (k) 2t – e^t (l) tan (z – 1 / 2) (m) e^t + e^-t (n) 3 sec (4x – 1) ...

(a) \int x^2+9 \mathrm{~d} x=\int x^2 \math...
Question: 13.3

## Use Table 13.1 to integrate the following functions: (a) x^4 (b) cos kx, where k is a constant (c) sin(3x + 2) (d) 5.9 (e) tan(6t − 4) (f) e^−3z (g) 1 / x² (h) cos 100nπt, where n is a constant ...

(a) From Table 13.1, we find  \int x^n \mat...
Question: 13.2

## Find  d / dx (x^n+1 / n + 1 + c) and hence deduce that  ∫ x^n dx = x^n + 1 / n + 1 + c. ...

From Table 10.1 we find \frac{\mathrm{d}}{\...