Question

How do you evaluate the integral of $\int { - 9x\cos (7x)dx} $?

Answer 1

Hint: First we will take out the constant term from the integral and then use the ILATE rule to integrate the multiple of algebraic and trigonometric functions inside.

Complete step-by-step answer:
We are given that we are required to evaluate the value of $\int { - 9x\cos (7x)dx} $.
Let us assume that this is equal to I.
So, we have: $I = \int { - 9x\cos (7x)dx} $
We can write this as: $I = - 9\int {x\cos (7x)dx} $ ……………(1)
Let us assume $J = \int {x\cos (7x)dx} $
Now, we have two functions inside the integral sign, one is x which is an algebraic function and another one is cos (7x) which is a trigonometric function.
Now, we also know that we have an ILATE rule according to which we will take x as the first function and cos (7x) as the second function.
Now, we will get:-
$ \Rightarrow J = x\int {\cos (7x)dx} - \int {\int {\left( {\cos (7x)dx} \right)} dx} $
Now, we know that the integration of cosine function is given by the following expression:-
$ \Rightarrow \int {\cos (ax)dx = \dfrac{{\sin ax}}{a}} $
Therefore, we will get the following expression:-
\[ \Rightarrow J = x\dfrac{{\sin (7x)}}{7} - \int {\dfrac{{\sin (7x)}}{7}dx} \]
We can write this as:-
\[ \Rightarrow J = x\dfrac{{\sin (7x)}}{7} - \dfrac{1}{7}\int {\sin (7x)dx} \]
Now, we know that the integration of sine function is given by the following expression:-
$ \Rightarrow \int {\sin (ax)dx = - \dfrac{{\cos ax}}{a}} $
So, we will get:-
\[ \Rightarrow J = \dfrac{{x\sin (7x)}}{7} + \dfrac{1}{7} \times \dfrac{{\cos (7x)}}{7} + C\]
On simplifying it, we will get:-
\[ \Rightarrow J = \dfrac{{x\sin (7x)}}{7} + \dfrac{{\cos (7x)}}{{49}} + C\]
Putting this in equation number (1), we will then obtain the following expression:-
\[ \Rightarrow I = - 9\left( {\dfrac{{x\sin (7x)}}{7} + \dfrac{{\cos (7x)}}{{49}} + C} \right)\]
We can write this as following expression:-

\[ \Rightarrow I = - \dfrac{{9x\sin (7x)}}{7} - \dfrac{{9\cos (7x)}}{{49}} + C'\]

Note:
The students must note that the ILATE rule we mentioned above states that we take the first function in respective order as ILATE, where I stands for inverse function, L stands for logarithmic function, A stands for algebraic function, T stands for trigonometric functions and E stands for exponential function.
Now, if f (x) is the first function and g (x) is the second function taken according to the ILATE rule, then we have:-
$ \Rightarrow \int {f(x).g(x)dx = f(x)\int {g(x)dx - \int {\left( {\dfrac{d}{{dx}}\left( {f(x)} \right)\int {g(x)dx} } \right)dx} } } $
The students must also not forget to put the constant in any indefinite integral as we did above as C’.