Question Statement

Evaluate the integral:

$\int x^4\sqrt{3x^5-5}dx$

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Background and Explanation

This integral is solved using the substitution method (u-substitution). Notice that the integrand contains a composite function $\sqrt{3x^5-5}$ and the remaining factor $x^4$ is proportional to the derivative of the inner function $3x^5-5$.

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Solution

Begin by rewriting the square root as a rational exponent to better identify the substitution:

$I=\int x^4\sqrt{3x^5-5}dx=\int {\left(3x^5-5\right)}^{\frac{1}{2}}\cdot x^{4}dx$

Step 1: Choose the substitution

Let $t=3x^5-5$. This is the inner function of the composite expression.

Step 2: Compute the differential

Differentiate both sides with respect to $x$:

$\frac{d}{dx}(3x^5-5)=\frac{dt}{dx}$

$15x^4=\frac{dt}{dx}$

Solving for $x^{4}dx$:

$15x^{4}dx=dt$

$x^{4}dx=\frac{1}{15}dt$

Step 3: Substitute into the integral

Replace $3x^5-5$ with $t$ and $x^{4}dx$ with $\frac{1}{15}dt$:

$I=\int t^{\frac{1}{2}}\cdot \frac{1}{15}dt$

Factor out the constant:

$I=\frac{1}{15}\int t^{\frac{1}{2}}dt$

Step 4: Apply the power rule

Using the power rule for integration $\int t^{n}dt=\frac{t^{n+1}}{n+1}+c$ where $n=\frac{1}{2}$:

$I=\frac{1}{15}\cdot \frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c$

$I=\frac{1}{15}\cdot \frac{t^{\frac{3}{2}}}{\frac{3}{2}}+c$

Step 5: Simplify the expression

Dividing by $\frac{3}{2}$ is equivalent to multiplying by $\frac{2}{3}$:

$I=\frac{1}{15}\cdot \frac{2}{3}{t}^{\frac{3}{2}}+c$

$I=\frac{2}{45}{t}^{\frac{3}{2}}+c$

Step 6: Substitute back to the original variable

Replace $t$ with $3x^5-5$:

$I=\frac{2}{45}{\left(3x^5-5\right)}^{\frac{3}{2}}+c$

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Key Formulas or Methods Used

• Substitution Rule: $\int f(g(x))\cdot g^{\prime }(x)dx=\int f(u)du$ where $u=g(x)$
• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$ (for $n\ne -1$)
• Differential Calculation: $dt=\frac{dt}{dx}\cdot dx$

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Summary of Steps

1. Rewrite the radical as a rational exponent: $(3x^5-5{)}^{1/2}$
2. Identify substitution: Let $t=3x^5-5$
3. Find differential: $dt=15x^{4}dx$, therefore $x^{4}dx=\frac{dt}{15}$
4. Transform the integral: $\frac{1}{15}\int t^{1/2}dt$
5. Integrate using power rule: $\frac{1}{15}\cdot \frac{2}{3}{t}^{3/2}+c$
6. Back-substitute: $\frac{2}{45}(3x^5-5{)}^{3/2}+c$