Question

How do you find the integral of \[\int {\cos \left( {2x + 1} \right)dx} \] using substitution ?

Answer 1

Hint:Integration is the process of finding the antiderivative. The integration of g′(x) with respect to dx is given by \[\int {{g^1}\left( x \right)dx = g\left( x \right) + C} \].Here C is the constant of integration and we can find the integral by using substitution i.e., differentiating with respect to x and solve for dx and find the integral of all the terms.

Complete step by step answer:
The given function is \[\int {\cos \left( {2x + 1} \right)dx} \].As we need to find the integral, let us rewrite the function as,
\[u = 2x + 1\]
Differentiate u with respect to x as
\[\dfrac{{du}}{{dx}} = 2\]
\[\Rightarrow du = 2dx\]
Let us solve for dx, we get
\[\dfrac{1}{2}du = dx\]
Use substitution method and factoring out the constant, we have
\[\dfrac{1}{2}\int {\cos u \cdot du} = \dfrac{1}{2}\sin u + C\]
Rewriting in terms of x, we get the integral as
\[\therefore\int {\cos \left( {2x + 1} \right)dx} = \dfrac{1}{2}\sin \left( {2x + 1} \right) + C\]

Hence,the integral of \[\int {\cos \left( {2x + 1} \right)dx} \] using substitution is $\dfrac{1}{2}\sin \left( {2x + 1} \right) + C$.

Note:There are different integration methods that are used to find an integral of some function, which is easier to evaluate the original integral. Hence, based on the function given we can find the integration of the function i.e., by using the integration methods as the details are given as additional information.