Question

section 3.5: substitution (homework)
score: 130/150 answered: 13/15
question 14
evaluate. use a capital ( c ) for any constant.
int 4 x e^{3 x^{2}-1} d x=

Explanation:

Step1: Let \(u = 3x^{2}-1\)

Differentiate \(u\) with respect to \(x\): \(du=6x dx\), then \(x dx=\frac{1}{6}du\)

Step2: Rewrite the integral

The integral \(\int4xe^{3x^{2}-1}dx\) becomes \(\int4e^{u}\cdot\frac{1}{6}du\)
Simplify the coefficient: \(\frac{4}{6}\int e^{u}du=\frac{2}{3}\int e^{u}du\)

Step3: Integrate \(e^{u}\)

Since \(\int e^{u}du = e^{u}+C\), we have \(\frac{2}{3}e^{u}+C\)

Step4: Substitute back \(u = 3x^{2}-1\)

We get \(\frac{2}{3}e^{3x^{2}-1}+C\)

Answer:

\(\frac{2}{3}e^{3x^{2}-1}+C\)