Evaluate ∫ (e^(log x) + sin x) cos x dx

Q: Why can't I just integrate $(e^{\log x} + \sin x)$ and $\cos x$ separately and multiply them?

That is a classic trap! Integration is not distributive over multiplication. Just as $\int f(x)g(x) dx \neq \int f(x) dx \cdot \int g(x) dx$, you must first expand the integrand into a sum of terms before you can integrate them individually. Always simplify the expression first!

Q: How do I know when to use Integration by Parts versus substitution?

Think of it as a hierarchy. If you see a product of two functions, check if one is the derivative of the other (Substitution). If not, and you have a mix of algebraic and trigonometric functions, use the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) to assign $u$ for Integration by Parts. Here, $x$ is Algebraic and $\cos x$ is Trigonometric, so IBP is the way to go.

Q: Why did we use $\sin 2x$ for the second part instead of just using substitution?

You absolutely could use substitution! If you let $u = \sin x$, then $du = \cos x dx$, and the integral becomes $\int u du = \frac{u^2}{2} = \frac{\sin^2 x}{2}$. However, using the double-angle identity $\sin 2x = 2 \sin x \cos x$ is often faster and keeps the answer in terms of $\cos 2x$, which is a standard form. Both methods are mathematically sound and will lead to the same result up to a constant.

Question

Evaluate $\int (e^{\log x} + \sin x) \cos x dx$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, my dear student, to a journey through the heart of calculus. Today, we are not just solving a problem; we are peeling back the layers of a mathematical mystery.

We are faced with the integral:

$$I = \int (e^{\log x} + \sin x) \cos x \, dx$$

At first glance, it might seem like a chaotic mess of functions, but in the JEE, chaos is often just order in disguise.

### The Logarithmic Trap

The first thing that catches the eye is $e^{\log x}$. Many students panic here, thinking they need to perform some complex substitution.

But remember the fundamental relationship between the exponential function and the natural logarithm: they are inverses. They undo each other.

Thus, $e^{\log x}$ is simply $x$. Just like that, the intimidating term vanishes, leaving us with a much friendlier expression:

$$I = \int (x + \sin x) \cos x \, dx$$

### The Art of Distribution

Now that we have simplified the core, we must address the structure. We have a sum inside a bracket, all multiplied by $\cos x$.

Integration is a linear operator, but it does not play well with products. We must distribute the $\cos x$ to break this product into a sum.

Applying the distributive property, our integrand transforms into $x \cos x + \sin x \cos x$. Now, we can split this into two distinct integrals:

$$I_1 = \int x \cos x \, dx \quad 	ext{and} \quad I_2 = \int \sin x \cos x \, dx$$

We have turned one big, scary problem into two smaller, manageable ones.

### The Duality of Integration

Let us tackle $I_1 = \int x \cos x \, dx$. We have an algebraic function $x$ multiplied by a trigonometric function $\cos x$.

This is the classic scenario for **Integration by Parts**. We invoke the ILATE rule, which tells us to choose $u$ as the algebraic function. So, we set $u = x$ and $dv = \cos x \, dx$.

Applying the formula $\int u \, dv = uv - \int v \, du$, we get:

$$I_1 = x \sin x - \int \sin x \, dx = x \sin x + \cos x$$

Now, for $I_2 = \int \sin x \cos x \, dx$. While we could use substitution, the double-angle identity $\sin 2x = 2 \sin x \cos x$ is a powerful tool in our arsenal.

We rewrite the integral as:

$$I_2 = \frac{1}{2} \int \sin 2x \, dx = \frac{1}{2} \left( -\frac{\cos 2x}{2} ight) = -\frac{1}{4} \cos 2x$$

### Final Synthesis

We have conquered both parts. Now, we bring them together in a grand finale.

Adding $I_1$ and $I_2$, we arrive at the final expression. We must never forget the constant of integration, $C$, which represents the family of all possible antiderivatives.

The **final result** is:

$$x \sin x + \cos x - \frac{1}{4} \cos 2x + C$$

You see? With patience, logic, and the right tools, even the most intimidating problems yield to the beauty of mathematics. Keep practicing, keep questioning, and most importantly, keep falling in love with the process.

Animated Solution for Mathematics - Indefinite Integration: Evaluate $\int (e^{l o g x} + sin x) cos x d x$

Visualized Solution

Simplify $e^{l o g x}$

Distribute $cos x$

Split the Integral

Setup Integration by Parts

Apply IBP Formula

Integrate $sin x$

Use $sin 2 x$ Identity

Integrate $sin 2 x$

Final Result

The Sigma Insight: Integration by Parts

Analyzing the Setup

The Logarithmic Trap

The Art of Distribution

The Duality of Integration

Final Synthesis

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Animated Solution for Mathematics - Indefinite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(elogx+sinx)cosxdx

Visualized Solution

Simplify .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } elogx

Distribute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cosx