Join the growing community of curious minds on IDNLearn.com and get the answers you need. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
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.
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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.