Key Concepts and Formulas

• Method of Substitution for Definite Integrals: If $u = g(x)$, then $\int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$. This rule allows us to simplify integrals by changing the variable of integration and adjusting the limits accordingly.
• Power Rule for Integration: $\int u^n du = \frac{u^{n+1}}{n+1} + C$, for $n \neq -1$. This is a fundamental rule for integrating polynomial and power functions.
• Algebraic Manipulation: The ability to factor and rearrange terms within an integrand is crucial for identifying suitable substitutions.

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Step-by-Step Solution

Step 1: Algebraic Manipulation of the Integrand

The given integral is I=\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x. To make the substitution method effective, we need to identify a part of the integrand whose derivative is related to the remaining part. Let's factor out $x^7$ from the first term: $x^{21}+x^{14}+x^{7} = x^7(x^{14}+x^{7}+1)$.

Now, consider how to bring the factor $x^7$ inside the parenthesis raised to the power of $1/7$. If we have a term $A \cdot B^{1/7}$, we can write it as $(A^7 \cdot B)^{1/7}$. However, it's often more useful to bring a factor into the term under the power by raising it to the appropriate power. Let's rewrite the integrand by factoring $x^7$ from the first term and then manipulate the expression to facilitate substitution. I=\int_\limits{0}^{1} x^{7}\left(x^{14}+x^{7}+1\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x This form doesn't immediately suggest a substitution. Let's try a different approach by factoring $x$ from the first term and seeing if we can manipulate it differently. Consider the first term: $x^{21}+x^{14}+x^{7}$. We can write this as $x \cdot (x^{20}+x^{13}+x^6)$. The integral becomes: I=\int_\limits{0}^{1} x\left(x^{20}+x^{13}+x^{6}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x To bring the factor $x$ inside the parenthesis $(2 x^{14}+3 x^{7}+6)^{1 / 7}$, we must raise it to the power of 7: $x \cdot (\text{expression})^{1/7} = (x^7 \cdot \text{expression})^{1/7}$. So, we can rewrite the integral as: I=\int_\limits{0}^{1}\left(x^{20}+x^{13}+x^{6}\right)\left(x^7 \cdot \left(2 x^{14}+3 x^{7}+6\right)\right)^{1 / 7} d x I=\int_\limits{0}^{1}\left(x^{20}+x^{13}+x^{6}\right)\left(2 x^{21}+3 x^{14}+6 x^{7}\right)^{1 / 7} d x This manipulation is key because the term $(x^{20}+x^{13}+x^{6})$ is proportional to the derivative of the expression inside the power $1/7$.

Step 2: Defining the Substitution

Let $u$ be the expression inside the seventh root: Let $u = 2x^{21}+3x^{14}+6x^{7}$.

Now, we find the differential $du$ by differentiating $u$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx}(2x^{21}+3x^{14}+6x^{7})$ $\frac{du}{dx} = 2(21x^{20}) + 3(14x^{13}) + 6(7x^{6})$ $\frac{du}{dx} = 42x^{20} + 42x^{13} + 42x^{6}$ $\frac{du}{dx} = 42(x^{20}+x^{13}+x^{6})$ Rearranging this, we get: $du = 42(x^{20}+x^{13}+x^{6}) dx$ This implies that $(x^{20}+x^{13}+x^{6}) dx = \frac{1}{42} du$. This is exactly the remaining part of our integrand, confirming the substitution is appropriate.

Step 3: Changing the Limits of Integration

Since we are dealing with a definite integral, we must change the limits of integration from $x$ to $u$. The original limits are $x=0$ and $x=1$.

• When $x=0$, the new lower limit for $u$ is: $u_{lower} = 2(0)^{21}+3(0)^{14}+6(0)^{7} = 0+0+0 = 0$.
• When $x=1$, the new upper limit for $u$ is: $u_{upper} = 2(1)^{21}+3(1)^{14}+6(1)^{7} = 2(1)+3(1)+6(1) = 2+3+6 = 11$.

Now, substitute $u$ and $du$ into the integral and change the limits: $I = \int_{0}^{11} u^{1/7} \left(\frac{1}{42} du\right)$ $I = \frac{1}{42} \int_{0}^{11} u^{1/7} du$

Step 4: Performing the Integration

We now integrate $u^{1/7}$ with respect to $u$ using the power rule: $\int u^n du = \frac{u^{n+1}}{n+1}$. Here, $n = 1/7$. So, $n+1 = 1/7 + 1 = 8/7$. $\int u^{1/7} du = \frac{u^{8/7}}{8/7} = \frac{7}{8} u^{8/7}$ Now, apply the definite integral with the changed limits: $I = \frac{1}{42} \left[\frac{7}{8} u^{8/7}\right]_{0}^{11}$ $I = \frac{1}{42} \left(\frac{7}{8} (11)^{8/7} - \frac{7}{8} (0)^{8/7}\right)$ $I = \frac{1}{42} \left(\frac{7}{8} (11)^{8/7} - 0\right)$ $I = \frac{1}{42} \cdot \frac{7}{8} (11)^{8/7}$

Simplify the constant factor: $I = \frac{7}{42 \cdot 8} (11)^{8/7}$ $I = \frac{1}{6 \cdot 8} (11)^{8/7}$ $I = \frac{1}{48} (11)^{8/7}$

Step 5: Comparing with the Given Form and Finding l, m, n

The problem states that the integral is equal to $\frac{1}{l}(11)^{m / n}$. Our calculated value is $I = \frac{1}{48} (11)^{8/7}$. By comparing these two forms, we have: $l = 48$ $m = 8$ $n = 7$

We are given that $l, m, n \in \mathbb{N}$ and $m$ and $n$ are coprime. $l=48$ is a natural number. $m=8$ and $n=7$ are natural numbers. To check if $m$ and $n$ are coprime, we find their greatest common divisor (GCD). The divisors of 8 are {1, 2, 4, 8} and the divisors of 7 are {1, 7}. The GCD(8, 7) = 1, so $m$ and $n$ are coprime.

Finally, we need to find the value of $l+m+n$. $l+m+n = 48 + 8 + 7 = 63$

Common Mistakes & Tips

• Incorrect Algebraic Manipulation: A common pitfall is misplacing factors inside or outside powers. Ensure that when a factor $x^k$ enters a term $(...)^{1/p}$, it becomes $(x^{kp} \cdot ...)^{1/p}$.
• Forgetting to Change Limits: For definite integrals, always remember to transform the limits of integration according to the substitution.
• Simplifying Constants Incorrectly: Double-check the arithmetic when simplifying fractions involving the constants from the integration and the substitution factor.
• Coprime Condition: Always verify that $m$ and $n$ are coprime as stated in the problem. If they are not, they must be reduced by dividing by their GCD.

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Summary

The problem was solved using the method of substitution for definite integrals. The integrand was first algebraically manipulated to reveal a suitable substitution. By letting $u = 2x^{21}+3x^{14}+6x^{7}$, its differential $du$ was found to be proportional to the remaining part of the integrand. The limits of integration were changed accordingly. After performing the integration using the power rule and simplifying the result, the integral was compared to the given form $\frac{1}{l}(11)^{m / n}$ to identify $l$, $m$, and $n$. The coprimality of $m$ and $n$ was verified, and the final sum $l+m+n$ was calculated.

The final answer is $\boxed{63}$.