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When do you integrate using substitution?
You integrate using substitution when you are presented with a function within a function. U-subs should be the first method you try when tackling an integration problem. -
An example of a function within a function
integral[sin(3x+5) dx] This is considered a "function within a function" because (3x+5) is a function of x and it is within a sine function. -
The first step
Now that we have identified the function within a function, let's call it u. So, u = 3x + 5. And du = 3 dx
We are left with integral[sin(u) dx]* *If the original intgral had limits of integration, drop them out now. They will be added back in later. -
Step 2
Before we can integrate integral[sin(u) dx] we need to replace the dx since the new integral is in terms of u. Since du = 3 dx, we know dx= (1/3) du. So we can change the integral to read: integral[(1/3)sin(u) du] -
Step 3
This is now an easy function to integrate. Constants come along for the ride so 1/3 can be pulled to the front and we can integrate sin(u) du to be cos(u) + C where C is any constant. -
Step 4
We are left with (1/3)cos(u) + C. We can now substitute our u = 3x + 5 back in. -
The final answer
Our final answer now reads (1/3)cos(3x+5) + C If we had original limits of integration, they can be subbed in at this point.