(1) replace x = tan t, -pi/2.

DX = sec 2 Tate

(2) radical sign (1+x 2) = radical sign (1+tan t 2) = sectarian integral.

Integral sec^3

Integral second t sec^2 t dt

= integration second t d (tan t)

(3) partial integration?

= sec t * tan t- integral tan t *

=sec t * tan t integral (sec2t- 1) sect dt.

= sec t * tan t- integral sec 3 dtt+ integral sectddt

(4) Both parties? Integral sec^3, merge to the left.

2 integration seconds 3 t dt = seconds t tan t+ln | seconds t+tan t |

(5) the integral sec3tdt =1/2 * [sect tantt+ln | sect+tantt |]+c.

(6) Then you have to go to a meeting, x=tan t, sec t= root number (1+tan 2t) = root number (1+x 2).

Integral = 1/2*[ x* radical sign (1+x 2)+ln | x+radical sign (1+x 2) |]+c

：

1, integral is a core concept in calculus and mathematical analysis. Usually divided into definite integral and indefinite integral. Intuitively speaking, for a given positive real function, the definite integral in the real number interval can be understood as the area value (a definite real value) of the curve trapezoid surrounded by curves, lines and axes on the coordinate plane.

2. The driving force of overall development comes from the demand in practical application. In practice, some unknowns can sometimes be roughly estimated, but with the development of science and technology, it is often necessary to know the exact values. If the area or volume of simple geometry is needed, the known formula can be applied. For example, the volume of a rectangular swimming pool can be calculated by length x width x height. But if the swimming pool is oval, parabolic or more irregular, it is necessary to calculate the volume by integral. In physics, it is often necessary to know the cumulative effect of one physical quantity (such as displacement) on another physical quantity (such as force), and integration is also needed at this time.

3. A function is said to be integrable if its integral exists and is finite. Generally speaking, the integrand function does not necessarily have only one variable, and the integral domain can also be a space with different dimensions, even an abstract space without intuitive geometric significance. As mentioned above, for a real function f with only one variable x, the integral of f in the closed interval [a, b] is written as.

4. Besides indicating that X is the variable to be integrated in F (integral variable), it can also indicate different meanings. In Riemann integral,? A mark representing a division interval; In Lebesgue integral, a measure is expressed; Or just an independent quantity (differential form). The integral on the general interval or integral interval j, j can be written as? .

5. If there is more than one variable, for example, in the double integral, the integral of the function in the region D is recorded as or.

6. Integral by parts is an important and basic method to calculate integral in calculus.

7. Its main principle is to use the differential formula of two multiplication functions to convert the required integral into the integral of another relatively simple function. According to the basic function types that constitute the integrand function, the order of partial integrals is arranged into a formula: "anti-power three fingers"

8. Refer to five basic functions respectively: inverse trigonometric function, logarithmic function, power function, trigonometric function and exponential function integration.

References:

Baidu Encyclopedia: Integral