In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable with others. Let’s consider an example for better understanding.
Simplify ∫ 3x2 sin (x3) dx.
Answer:
Let I = ∫ 3x2 sin (x3) dx.
In order to evaluate the given integral lets substitute any variable by a new variable as:
Let x3 be t for the given integral.
Then, dt = 3x2 dx
Therefore,
I = ∫ 3x2 sin (x3) dx = ∫ sin (x3) (3x2 dx)
Now, substitute t for x3 and dt for 3x2 dx in the above integral.
I = ∫ sin (t) (dt)
As ∫ sin x dx = -cos x + C, thus
I = -cos t + C
Again, substitute back x3 for t in the expression as:
I = ∫ 3x2 sin (x3) dx = -cos x3 + C
Which is the required integral. |
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