Animated Solution for Mathematics - Indefinite Integration: The integral $\int \cos ({\log }_{e}x)dx$ is equal to : (where C is a constant of integration)

Welcome, future engineers! Today, we are going to peel back the layers of a truly elegant integral: $\int \cos ({\log }_{e}x)dx$.

At first glance, this problem might seem intimidating. We have a trigonometric function, cosine, but its argument is not a simple variable; it is a logarithm.

This is a classic scenario in JEE Advanced mathematics where the structure of the function is designed to test your ability to transform and simplify.

The Art of Substitution

When you encounter a function nested inside another, like $\cos ({\log }_{e}x)$, the most powerful tool in your arsenal is substitution. We want to simplify the argument of the cosine function.

Let us introduce a new variable, $t$, such that $t={\log }_{e}x$. This small, deliberate choice is the key that unlocks the entire problem.

By defining $t={\log }_{e}x$, we are essentially asking: "What if we viewed this function from the perspective of the logarithm?"

The Transformation

Now, we must be careful. We cannot just replace ${\log }_{e}x$ with $t$ and leave the $dx$ as it is. We need to express $x$ and $dx$ in terms of $t$.

From our substitution $t={\log }_{e}x$, we can easily convert this to the exponential form: $x=e^t$.

Now, let us find the differential $dx$. By differentiating both sides with respect to $t$, we get:

This is the "price" of our transformation, but it is a price well worth paying. Our integral, which was once $\int \cos ({\log }_{e}x)dx$, now transforms into the much more manageable form:

The Power of the Standard Form

Look at what we have achieved! We have transformed a complex logarithmic integral into the standard form $\int e^t\cos (t)dt$.

This is a famous integral that often appears in competitive exams. While you could solve this using integration by parts twice—a process that is perfectly valid but time-consuming—we can use a powerful standard formula.

The integral of the form $\int e^{at}\cos (bt)dt$ is given by:

By comparing our integral $\int e^t\cos (t)dt$ with this formula, we can easily identify our constants: $a=1$ and $b=1$.

Substituting these values into the formula, we get:

Simplifying this, we arrive at $I=\frac{e^t}{2}[\cos (t)+\sin (t)]+C$.

The Final Reveal

We are almost at the finish line, but we must remember that our original problem was in terms of $x$. We cannot leave our answer in terms of $t$.

We must perform the back-substitution. Recall that $t={\log }_{e}x$ and $e^t=x$.

Replacing these back into our expression, we get the final, elegant result:

And there you have it! We started with a daunting expression and, through the power of substitution and standard formulas, arrived at a clean, beautiful solution.

Remember, in JEE Advanced, the goal is not just to solve the problem, but to see the underlying structure. Keep practicing, stay curious, and never let a complex-looking integral intimidate you. You have the tools to break it down!