To solve the integral [tex]\(-\int \sqrt[3]{7x - 10} \, dx\)[/tex], follow these steps:
1. Identify the integrand: We need to find the integral of [tex]\(- (7x - 10)^{1/3}\)[/tex].
[tex]\[
\int - (7x - 10)^{1/3} \, dx
\][/tex]
2. Simplify the integrand: Rewrite the integrand for simplicity.
[tex]\[
\int - (7x - 10)^{\frac{1}{3}} \, dx
\][/tex]
The negative sign can be taken out of the integral:
[tex]\[
-\int (7x - 10)^{\frac{1}{3}} \, dx
\][/tex]
3. Substitute and simplify: Use a substitution to simplify the integration process.
Let [tex]\( u = 7x - 10 \)[/tex]. Then, [tex]\( du = 7 \, dx \)[/tex] or [tex]\( dx = \frac{1}{7} \, du \)[/tex].
4. Rewrite the integral in terms of u:
[tex]\[
- \int (u)^{\frac{1}{3}} \cdot \frac{1}{7} \, du
\][/tex]
This simplifies to:
[tex]\[
- \frac{1}{7} \int u^{\frac{1}{3}} \, du
\][/tex]
5. Integrate: Now integrate [tex]\( u^{\frac{1}{3}} \)[/tex] with respect to [tex]\( u \)[/tex].
The integral of [tex]\( u^{\frac{1}{3}} \)[/tex] is obtained by increasing the exponent by 1 and then dividing by the new exponent:
[tex]\[
\int u^{\frac{1}{3}} \, du = \frac{u^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4} u^{\frac{4}{3}}
\][/tex]
6. Combine and simplify: Substitute back and combine the constant multiples.
[tex]\[
- \frac{1}{7} \cdot \frac{3}{4} u^{\frac{4}{3}} = - \frac{3}{28} u^{\frac{4}{3}}
\][/tex]
Replace [tex]\( u \)[/tex] with [tex]\( 7x - 10 \)[/tex]:
[tex]\[
- \frac{3}{28} (7x - 10)^{\frac{4}{3}}
\][/tex]
Thus, the integral [tex]\(-\int \sqrt[3]{7x - 10} \, dx\)[/tex] results in:
[tex]\[
-0.107142857142857 \cdot (7x - 10)^{\frac{4}{3}}
\][/tex]
where [tex]\(-0.107142857142857\)[/tex] is approximately [tex]\(-\frac{3}{28}\)[/tex], and this completes the integration process.
