Question

question 27
find the indefinite integral: \\( \int 3x(5x^2 + 3)^7 dx \\).
\\( \frac{3}{80}(5x^2 + 3)^8 + c \\)
no correct answer choice is given.
\\( x^3(5x^2 + 3)^8 + c \\)
\\( \frac{3}{2}(5x^2 + 3)^8 + c \\)
\\( 3x^2(5x^2 + 3)^8 + c \\)

Explanation:

Step1: Use Substitution Method

Let \( u = 5x^2 + 3 \), then find \( du \). Differentiating \( u \) with respect to \( x \), we get \( du = 10x \, dx \), so \( x \, dx=\frac{du}{10} \).

Step2: Rewrite the Integral

The integral \( \int 3x(5x^2 + 3)^7 dx \) can be rewritten as \( 3\int (5x^2 + 3)^7 \cdot x \, dx \). Substituting \( u = 5x^2 + 3 \) and \( x \, dx=\frac{du}{10} \), we have \( 3\int u^7 \cdot \frac{du}{10}=\frac{3}{10}\int u^7 du \).

Step3: Integrate with Respect to \( u \)

Using the power rule for integration \( \int u^n du=\frac{u^{n + 1}}{n+1}+C \) (where \( n = 7 \)), we get \( \frac{3}{10}\cdot\frac{u^{8}}{8}+C=\frac{3}{80}u^{8}+C \).

Step4: Substitute Back \( u \)

Substitute back \( u = 5x^2 + 3 \), so the integral is \( \frac{3}{80}(5x^2 + 3)^8 + C \).

Answer:

\(\frac{3}{80}(5x^2 + 3)^8 + C\) (the first option: \(\boldsymbol{\frac{3}{80}(5x^2 + 3)^8 + C}\))