Q.Integrate the following function: $\int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx$ equals (A) $10^x - x^{10} + C$ (B) $10^x + x^{10} + C$ (C) $(10^x - x^{10})^{-1} + C$ (D) $\log (10^x + x^{10}) + C$

Concept: U Substitution — the numerator is exactly the derivative of the denominator.

Let

$u = x^{10} + 10^x$.

Then

$\frac{du}{dx} = 10x^9 + 10^x \log_e 10$,

so $du = (10x^9 + 10^x \log_e 10)\, dx$.

The integral becomes …

The key to this problem is recognising a pattern that appears again and again in integration: when you see a fraction where the numerator looks like the derivative of the denominator, you're looking at a natural logarithm result.

Let’s check that idea. If you have $\int \frac{f'(x)}{f(x)} dx$, the answer is $\log |f(x)| + C$. Why? Because the derivative of $\log f(x)$ is $\frac{f'(x)}{f(x)}$ by the chain rule. So the whole game is: can we spot an $f(x)$ whose derivative matches the numerator?

Here the denominator is $x^{10} + 10^x$. Let’s differentiate it:

1. The derivative of $x^{10}$ is $10x^9$.
2. The derivative of $10^x$ is $10^x \log_e 10$ (since $\frac{d}{dx} a^x = a^x \log a$).

So $\frac{d}{dx}(x^{10} + 10^x) = 10x^9 + 10^x \log_e 10$.

That is exactly the numerator. So we have:

where $f(x) = x^{10} + 10^x$.

Therefore:

Since $x^{10} + 10^x$ is always positive for real $x$, we can drop the absolute value and write $\log (x^{10} + 10^x) + C$.

Method: Numerator is the derivative of the denominator $\Rightarrow$ logarithm

Use this whenever an integrand is a single fraction and the top looks like the derivative of the bottom. The antiderivative is then a natural logarithm — no formal substitution needed once you recognise it.

Steps

Step 1: Differentiate the denominator mentally.

Call the denominator $f(x)$ and compute $f'(x)$. Remember the non-polynomial derivatives that show up here: $\dfrac{d}{dx}a^x=a^x\log_e a$ and $\dfrac{d}{dx}x^n=nx^{n-1}$.

Step 2: Check whether $f'(x)$ equals the numerator.

If the numerator is exactly $f'(x)$ (or a constant multiple of it), the integrand is in the form $\dfrac{f'(x)}{f(x)}$.

Step 3: Apply the standard result. …

Common Mistakes

Mistake 1: Choosing option (B), the antiderivative of the numerator alone.

Why it's wrong: $10^x+x^{10}$ differentiates to the numerator, but this is a fraction $\dfrac{f'(x)}{f(x)}$, not just $f'(x)$ — the denominator changes the antiderivative into a logarithm. Correct approach: apply $\displaystyle\int\frac{f'(x)}{f(x)}\,dx=\log|f(x)|+C$.

Mistake 2: Using $\dfrac{d}{dx}10^x=10^x$ instead of $10^x\log_e 10$. …