Question

\(\int \sqrt{x}\sec \left(x^{\frac{3}{2}}\right)\tan \left(x^{\frac{3}{2}}\right)dx=\)

• A. \(\frac{2}{3}\sin \left(x^{\frac{3}{2}}\right)+c\)
• B. \(\frac{2}{3}\tan \left(x^{\frac{3}{2}}\right)+c\)
• C. \(\frac{2}{3}\cot \left(x^{\frac{3}{2}}\right)+c\)
• D. \(\frac{2}{3}\sec \left(x^{\frac{3}{2}}\right)+c\)

Hint:

Substitute u = x^(3/2) so that sqrt(x) dx becomes a multiple of du.

Answer 1

The Correct Option is D

Step 1: Check by differentiation
Differentiate $\dfrac{2}{3}\sec\left(x^{3/2}\right)$ using the chain rule.

Step 2: Calculation
$\dfrac{2}{3}\sec\left(x^{3/2}\right)\tan\left(x^{3/2}\right)\cdot\dfrac{3}{2}x^{1/2}=\sqrt{x}\sec\left(x^{3/2}\right)\tan\left(x^{3/2}\right)$.

Step 3: Conclusion
This equals the integrand, so the antiderivative is the fourth option.

Final Answer:
Differentiating two thirds sec of x to the three halves returns the integrand. \[ \boxed{\text{(D) }\dfrac{2}{3}\sec \left(x^{\frac{3}{2}}\right)+c} \]