Question
(4) \( \int \frac{d x}{(3 x+5)^{4}} \)
Ask by Chadwick Parsons. in India
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\int \frac{dx}{(3x + 5)^4} = -\frac{1}{9(3x + 5)^3} + C
\]
Solution
To evaluate the integral:
\[
\int \frac{dx}{(3x + 5)^4}
\]
**Step 1: Substitution**
Let’s make a substitution to simplify the integral.
Let:
\[
u = 3x + 5
\]
Then, compute the differential \( du \):
\[
\frac{du}{dx} = 3 \quad \Rightarrow \quad dx = \frac{du}{3}
\]
**Step 2: Rewrite the Integral in Terms of \( u \)**
Substitute \( u \) and \( dx \) into the original integral:
\[
\int \frac{dx}{(3x + 5)^4} = \int \frac{\frac{du}{3}}{u^4} = \frac{1}{3} \int u^{-4} \, du
\]
**Step 3: Integrate**
Integrate \( u^{-4} \) with respect to \( u \):
\[
\frac{1}{3} \int u^{-4} \, du = \frac{1}{3} \left( \frac{u^{-3}}{-3} \right) + C = -\frac{1}{9} u^{-3} + C
\]
**Step 4: Substitute Back to \( x \)**
Replace \( u \) with the original expression \( 3x + 5 \):
\[
-\frac{1}{9} u^{-3} + C = -\frac{1}{9(3x + 5)^3} + C
\]
**Final Answer:**
\[
\int \frac{dx}{(3x + 5)^4} = -\frac{1}{9(3x + 5)^3} + C
\]
where \( C \) is the constant of integration.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve the integral \( \int \frac{d x}{(3 x+5)^{4}} \), we can use the substitution method. Let \( u = 3x + 5 \), then \( du = 3dx \) or \( dx = \frac{du}{3} \). The integral transforms into: \[ \int \frac{1}{u^4} \cdot \frac{du}{3} = \frac{1}{3} \int u^{-4} \, du \] Now, integrating \( u^{-4} \) gives us: \[ \frac{1}{3} \cdot \left( -\frac{1}{3} u^{-3} \right) + C = -\frac{1}{9} u^{-3} + C \] Substituting back \( u = 3x + 5 \): \[ \int \frac{d x}{(3 x+5)^{4}} = -\frac{1}{9 (3x + 5)^{3}} + C \]
