Pure definite integration

3. Find, using calculus and showing each step of your working, $$\int _ { 1 } ^ { 4 } \left( 6 x - 3 - \frac { 2 } { \sqrt { x } } \right) \mathrm { d } x$$

1. Use calculus to find the value of

$$\int _ { 1 } ^ { 4 } ( 2 x + 3 \sqrt { } x ) d x$$

6

1. 1. Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
   2. Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
2. Find \(\int \frac { 6 } { x ^ { 3 } } d x\)

2 Find \(\int _ { 1 } ^ { 2 } \left( 12 x ^ { 5 } + 5 \right) \mathrm { d } x\).

6 Evaluate \(\int _ { 1 } ^ { 2 } \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

1 Find \(\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x\).

1. Evaluate

$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$

1. Evaluate

$$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } \mathrm {~d} x .$$

1. Evaluate

$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$

2 It is given that $$\int _ { 0 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 20 \text { and } \int _ { 3 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = - 10$$ Find the value of \(\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\) Circle your answer. \(- 30 - 101030\)

5

1. Find \(\int \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).
2. Hence find \(\int _ { 1 } ^ { 3 } \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).

Showing all necessary working, find \(\int_1^4 \left(\sqrt{x} + \frac{2}{\sqrt{x}}\right) \text{d}x\). [4]

Find \(\int_1^2 \left( x^4 - \frac{3}{x^2} + 1 \right) dx\), showing your working. [5]

Find \(\int_{2}^{5} \left(1 - \frac{6}{x^3}\right) dx\). [4]

Find \(\int_2^5 (2x^3 + 3) dx\). [3]

Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]

Find \(\int_1^2 (12x^5 + 5) dx\). [4]

Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]

Given that $$\int_0^{10} f(x) \, dx = 7$$ deduce the value of $$\int_0^{10} \left( f(x) + 1 \right) dx$$ Circle your answer. [1 mark] \(-3\) \quad \(7\) \quad \(8\) \quad \(17\)

Find the value of the integral: $$\int_0^1 \frac{x^{\frac{1}{2}} + x^{-\frac{1}{3}}}{x} \, dx$$ [4]

Find the exact value of $$\int_1^4 \left(10x^2 - 3x^2\right) dx.$$ [3]