Find the integrals of the function: \(sin^3 (2x+1)\)
Solution and Explanation
Let \(I = ∫sin^3(2x+1)\)
\(⇒ ∫sin^3(2x+1)dx = ∫sin^2(2x+1).sin(2x+1)dx\)
\(= ∫(1-cos^2(2x+1)sin(2x+1)dx\)
Let \(cos(2x+1)=t\)
\(⇒ -2sin(2x+1)dx = dt\)
\(⇒ sin(2x+1)dx = \frac{-dt}{2}\)
\(⇒ I = \frac{-1}{2} ∫(1-t^2)dt\)
\(=-\frac{1}{2}[t-\frac{t^3}{3}]\)
\(=\frac{-1}{2}{cos(2x+1)-\frac{cos^3(2x+1)}{3}}\)
\(=\frac{-cos(2x+1)}{2}+\frac{cos^3(2x+1)}{6}+C\)
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- Literature
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- Debentures
Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C