Question

The area under the curve \[ f(x)=x^5e^{x^3} \] between \(x=1\) and \(x=2\) is

• A. \(\dfrac{1}{3}e^8\)
• B. \(\dfrac{4}{3}e^8\)
• C. \(\dfrac{5}{3}e^8\)
• D. \(\dfrac{7}{3}e^8\)

Hint:

For integrals involving expressions like \(x^n e^{x^m}\), substitution using the exponent term often simplifies the integration.

Answer 1

The Correct Option is B

Step 1: Write the required integral.
The area under the curve is
\[ \int_1^2 x^5e^{x^3}\,dx \]

Step 2: Use substitution.
Let
\[ u=x^3 \] Then,
\[ \frac{du}{dx}=3x^2 \] \[ du=3x^2dx \] So,
\[ x^2dx=\frac{du}{3} \]
Now rewrite the integral:
\[ x^5dx=x^3(x^2dx) \] Since
\[ x^3=u, \] we get
\[ x^5dx=u\frac{du}{3} \]

Step 3: Change the limits.
When
\[ x=1,\quad u=1^3=1 \] When
\[ x=2,\quad u=2^3=8 \]
Thus, the integral becomes
\[ \int_1^8 \frac{u}{3}e^u\,du \] \[ =\frac{1}{3}\int_1^8 ue^u\,du \]

Step 4: Integrate by parts.
We know that
\[ \int ue^u\,du=e^u(u-1) \]
Therefore,
\[ \frac{1}{3}\left[e^u(u-1)\right]_1^8 \]

Step 5: Substitute the limits.
\[ =\frac{1}{3}\left[e^8(8-1)-e^1(1-1)\right] \] \[ =\frac{1}{3}\left[7e^8-0\right] \] \[ =\frac{7}{3}e^8 \]

Step 6: Final conclusion.
Hence, the area under the curve is
\[ \boxed{\frac{7}{3}e^8} \]
Therefore, the correct option is (D).