2. gxc2x2+3 dx Selast & du 니 get x in terms of u ut 2x2 +3 4-3 = ㄹ x² 4-3 x = 이들 4-3 z soke har x 4:2x2+3 du = 4x dx du =qy dx ax: 심 수 (m) 혜수 du du V2 + [(4-3)^² ^"] 나를) +[(*)* - (³›*]. · -(를)].
2. gxc2x2+3 dx Selast & du 니 get x in terms of u ut 2x2 +3 4-3 = ㄹ x² 4-3 x = 이들 4-3 z soke har x 4:2x2+3 du = 4x dx du =qy dx ax: 심 수 (m) 혜수 du du V2 + [(4-3)^² ^"] 나를) +[(*)* - (³›*]. · -(를)].
2. gxc2x2+3 dx Selast & du 니 get x in terms of u ut 2x2 +3 4-3 = ㄹ x² 4-3 x = 이들 4-3 z soke har x 4:2x2+3 du = 4x dx du =qy dx ax: 심 수 (m) 혜수 du du V2 + [(4-3)^² ^"] 나를) +[(*)* - (³›*]. · -(를)].
I need help with this integration by substituting problem.
Transcribed Image Text:## Integration by Substitution
### Problem 2:
Evaluate the integral of \(x(2x^2 + 3)^4 dx\).
### Solution:
1. **Substitute**:
- Let \(u = 2x^2 + 3\)
- Then \(du = 4x \, dx\)
- Therefore, \(dx = \frac{1}{4x} \, du\)
2. **Rewrite Integral**:
- Substitute \(u\) and \(dx\) into the original integral:
\[
\int x(2x^2 + 3)^4 dx = \int x u^4 \frac{1}{4x} \, du
\]
The \(x\) terms cancel out, giving:
\[
= \frac{1}{4} \int u^4 du
\]
3. **Integrate**:
- Integrate \(u^4\):
\[
\frac{1}{4} \int u^4 du = \frac{1}{4} \cdot \frac{u^5}{5} = \frac{1}{20} u^5 + C
\]
4. **Resubstitute** \(u\):
- Recall \(u = 2x^2 + 3\):
\[
\frac{1}{20} (2x^2 + 3)^5 + C
\]
Therefore, the final answer is:
\[
\int x (2x^2 + 3)^4 dx = \frac{1}{20} (2x^2 + 3)^5 + C
\]
### Diagram/Graph Explanation:
- The diagram illustrates the use of substitution for integration.
- The substitution \( u = 2x^2 + 3 \) is highlighted, showing how to transform the integral.
- Conversion steps involve isolating \( x \) in terms of \( u \), and subsequently replacing \( dx \) with \(\frac{1}{4x} du\).
- The integral is simplified by cancelling out \( x \) terms, integrating \( u \), and then substituting back to \( x \).
This method effectively handles more complex integrals by simplifying them through substitution.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![## Integration by Substitution
### Problem 2:
Evaluate the integral of \(x(2x^2 + 3)^4 dx\).
### Solution:
1. **Substitute**:
- Let \(u = 2x^2 + 3\)
- Then \(du = 4x \, dx\)
- Therefore, \(dx = \frac{1}{4x} \, du\)
2. **Rewrite Integral**:
- Substitute \(u\) and \(dx\) into the original integral:
\[
\int x(2x^2 + 3)^4 dx = \int x u^4 \frac{1}{4x} \, du
\]
The \(x\) terms cancel out, giving:
\[
= \frac{1}{4} \int u^4 du
\]
3. **Integrate**:
- Integrate \(u^4\):
\[
\frac{1}{4} \int u^4 du = \frac{1}{4} \cdot \frac{u^5}{5} = \frac{1}{20} u^5 + C
\]
4. **Resubstitute** \(u\):
- Recall \(u = 2x^2 + 3\):
\[
\frac{1}{20} (2x^2 + 3)^5 + C
\]
Therefore, the final answer is:
\[
\int x (2x^2 + 3)^4 dx = \frac{1}{20} (2x^2 + 3)^5 + C
\]
### Diagram/Graph Explanation:
- The diagram illustrates the use of substitution for integration.
- The substitution \( u = 2x^2 + 3 \) is highlighted, showing how to transform the integral.
- Conversion steps involve isolating \( x \) in terms of \( u \), and subsequently replacing \( dx \) with \(\frac{1}{4x} du\).
- The integral is simplified by cancelling out \( x \) terms, integrating \( u \), and then substituting back to \( x \).
This method effectively handles more complex integrals by simplifying them through substitution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F61f775f8-453f-4923-afa7-e7c3c948c07b%2Fz4pamor_processed.jpeg&w=3840&q=75)
