Question

Answer with explanation...​

Answer 1

Question:

$Find \: the \: value \: of \: \displaystyle \int\limits_{ - 2012}^{2012}(sin {x}^{3} + {x}^{5} + 1)dx$

Answer:

4024

Solution:

Please refer to the attachment.

Some properties of definite integrals:

$1. \: \displaystyle \int\limits_{a}^{b} f(x)dx = \displaystyle \int\limits_{a}^{b}f(z)dz$

$2. \: \displaystyle \int\limits_{a}^{b} f(x)dx = - \displaystyle \int\limits_{b}^{a}f(x)dx$

$3. \: \displaystyle \int\limits_{a}^{b} f(x)dx = \displaystyle \int\limits_{a}^{c}f(x)dx + \displaystyle \int\limits_{c}^{b}f(x)dx \\ (where \: a < c < b)$

$4. \: \displaystyle \int\limits_{a}^{b} f(x)dx = \displaystyle \int\limits_{a}^{b}f(a + b + x)dx$

$5. \: \displaystyle \int\limits_{0}^{a} f(x)dx = \displaystyle \int\limits_{0}^{a}f(a - x)dx$

$6. \: \displaystyle \int\limits_{ - a}^{a} f(x)dx =2 \displaystyle \int\limits_{0}^{a}f(x)dx \\ (when \: f(x) \: is \: an \: even \: function)$

$7. \: \displaystyle \int\limits_{ - a}^{a} f(x)dx = 0 \\ (when \: f(x) \: is \: an \: odd \: function)$

$8. \: \displaystyle \int\limits_{a}^{a} f(x)dx = 0$

$9. \: \displaystyle \int\limits_{0}^{2a} f(x)dx = \displaystyle \int\limits_{0}^{a}f(x)dx + \displaystyle \int\limits_{0}^{a}f(2a - x)dx$

$10. \: \displaystyle \int\limits_{0}^{2a} f(x)dx = 2\displaystyle \int\limits_{0}^{a}f(x)dx \\ (when \: f(2a - x)dx = f(x))$

$11. \: \displaystyle \int\limits_{0}^{2a} f(x)dx = 0 \\ (when \: f(2a - x)dx = -f(x))$

Odd and even function:

Odd function:

A function f(x) is said to be an odd function if , f(-x) = -f(x) .

Even function:

A function f(x) is said to be an even function if , f(-x) = f(x) .