Why is integration so much harder than differentiation? If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in terms of the integral of the products, and forget about composition! Why is this?
Here is an extremely generic answer. Differentiation is a "local" operation: to compute the derivative of a function at a point you only have to know how it behaves in a neighborhood of that point. But integration is a "global" operation: to compute the definite integral of a function in an interval you have to know how it behaves on the entire interval (and to compute the indefinite integral you have to know how it behaves on all intervals). That is a lot of information to summarize. Generally, local things are much easier than global things.

On the other hand, if you can do the global things, they tend to be useful because of how much information goes into them. That's why theorems like the fundamental theorem of calculus, the full form of Stokes' theorem, and the main theorems of complex analysis are so powerful: they let us calculate global things in terms of slightly less global things.
I will try to bring this to you in another way .Let us start by thinking in terms of something as simple as a straight line .
If I give you the equation of a line y = mx + c , it's slope can be easily determined which in this case is nothing but m .
.Now let me make the question a bit trickier .Let me say that the line given above intersects the x and y axis at some points .I ask you to give me the area between the line,the abcissa and the ordinate 

This is obviously not as easy as finding the slope .You  shall have to find the intersection of the line with the axis and get two points of intersection and then taking the origin as a third point find the area .
This  is not the only method of finding the area as we know there are loads of formulas for finding the area of a triangle .
Let us now view this in terms of curves .If the simple process of finding the slope in case of a line is translated to curves we get differential calculus which is a bit more complicated than the method of finding slopes of straight lines .

Add  finding the area under the curve to that and you get integral calculus which by our experience from straight lines we know should be much harder than finding the slope ie differentiation .Also  there is no one fixed method for finding the area of a figure .hence the Many methods of. Integration.