Integrate \[\int {\dfrac{{\sec x\tan x}}{{3\sec x + 5}}} dx\]

Hint: Integration is the calculation of an integral. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. When we speak about integrals, it is usually related to definite integrals. The indefinite integrals are used for antiderivatives. Integration is one of the two major calculus topics in Mathematics, apart from differentiation(which measures the rate of change of any function with respect to its variables).

Complete step-by-step answer:
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus .Derivatives can be generalized to functions of several real variables. The process of finding a derivative is called differentiation. The reverse process is called anti differentiation.
Formula used in the solution part:
\[\dfrac{{dy}}{{dx}}(\sec x) = \sec x\tan x\]
We have to find \[\int {\dfrac{{\sec x\tan x}}{{3\sec x + 5}}} dx\]
We will do it by substitution method.
Let \[3\sec x + 5 = z\]
Now differentiating both sides with respect to \[x\] we get ,
\[3\sec x\tan x dx = dz\]
Therefore we get ,
\[\sec x\tan x dx = \dfrac{{dz}}{3}\]
Therefore our integral becomes
\[\int {\dfrac{{\sec x\tan x}}{{3\sec x + 5}}} dx = \int {\dfrac{{dz}}{{3z}}} \]
\[ = \dfrac{{\log z}}{3} + C\]
Where C is the integration constant.
Now on substituting the value of \[z\] we get ,
\[\int {\dfrac{{\sec x\tan x}}{{3\sec x + 5}}} dx = \dfrac{1}{3}\log \left| {3\sec x + 5} \right| + C\]

Note: : The integration denotes the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly. In calculus, the concept of differentiating a function and integrating a function is linked using the theorem called the Fundamental Theorem of Calculus.