Answer :
To find the indefinite integral [tex]\int 5x^2 e^{-7x^3} \; dx[/tex], we can use the technique of substitution.
Choose a substitution: Let [tex]u = -7x^3[/tex]. Thus, the differential [tex]du = -21x^2 dx[/tex].
Solve for [tex]dx[/tex]: We can express [tex]dx[/tex] in terms of [tex]du[/tex] by rearranging the equation:
[tex]dx = \frac{du}{-21x^2}[/tex]Substitute into the integral: Replace [tex]u[/tex] and [tex]dx[/tex] in the integral:
[tex]\int 5x^2 e^{-7x^3} \; dx = \int 5x^2 e^u \cdot \frac{du}{-21x^2}[/tex]The [tex]x^2[/tex] terms cancel out, leading to:
[tex]\int \frac{5}{-21} e^u \; du = -\frac{5}{21} \int e^u \; du[/tex]Integrate: The integral of [tex]e^u[/tex] is [tex]e^u[/tex], so:
[tex]-\frac{5}{21} \int e^u \; du = -\frac{5}{21} e^u + C[/tex]Substitute back: Replace [tex]u[/tex] with [tex]-7x^3[/tex]:
[tex]-\frac{5}{21} e^{-7x^3} + C[/tex]
Thus, the indefinite integral [tex]\int 5x^2 e^{-7x^3} \; dx[/tex] is [tex]-\frac{5}{21}e^{-7x^3} + C[/tex].
From the given multiple-choice options, the correct answer is B. [tex]-\frac{5}{21}e^{-7x^3} + C[/tex].
