Integral substitusi $$ \int5x\left(7x^2-3\right)^6\differentialD x $$

To solve the integral 

\[
\int 5x \left(7x^2 - 3\right)^6 \, dx,
\]

we can use the substitution method. Let's set 

\[
u = 7x^2 - 3.
\]

Now, we need to find $du$:

\[
du = \frac{d}{dx}(7x^2 - 3) \, dx = 14x \, dx.
\]

We can solve for $dx$:

\[
dx = \frac{du}{14x}.
\]

Now we can rewrite the integral in terms of $u$:

1. Substitute $u$ for $7x^2 - 3$.
2. Since we have $5x \, dx$, we can express it using $du$:

\[
5x \, dx = \frac{5}{14} du.
\]

Thus, our integral becomes:

\[
\int 5x \left(7x^2 - 3\right)^6 \, dx = \int \frac{5}{14} u^6 \, du.
\]

Now, we can integrate:

\[
\int u^6 \, du = \frac{u^7}{7} + C.
\]

So,

\[
\int \frac{5}{14} u^6 \, du = \frac{5}{14} \cdot \frac{u^7}{7} + C = \frac{5}{98} u^7 + C.
\]

Finally, we substitute back $u = 7x^2 - 3$:

\[
\frac{5}{98} (7x^2 - 3)^7 + C.
\]

Thus, the solution to the integral is:

\[
\int 5x \left(7x^2 - 3\right)^6 \, dx = \frac{5}{98} (7x^2 - 3)^7 + C.
\]

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. What other techniques can be used for integrals involving polynomials?
2. Can you explain the process of finding $du$ in more detail?
3. What if the integral had a different polynomial in the parentheses?
4. How would you apply integration by parts to a similar problem?
5. What are some real-world applications of integral calculus? 

**Tip:** Always check if a substitution can simplify the integral before proceeding!

Discover how to solve the integral \int 5x(7x^2 - 3)^6 \, dx using substitution. This comprehensive guide covers key calculus concepts, including the substitution method and the integration of polynomial functions. Ideal for advanced students in Grades 11-12.

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