JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

$\int x^{2} sin x^{3} d x =$

Choose the correct answer:

Explanation

Step 1: Choose a substitution

Let $u = x^3$ .

Step 2: Find the derivative of $u$

Differentiating $u$ with respect to $x$ :

\frac{du}{dx} = 3x^2

du = 3x^2 \, dx

Step 3: Adjust the integral

We can rewrite the expression for $du$ to match the $x^2 \, dx$ term in our integral:

\frac{1}{3} du = x^2 \, dx

Step 4: Substitute into the integral

Now substitute $u$ and $\frac{1}{3} du$ into the original integral:

\int x^2 \sin x^3 \, dx = \int \sin(u) \cdot \frac{1}{3} \, du

\int x^2 \sin x^3 \, dx = \frac{1}{3} \int \sin u \, du

Step 5: Integrate

The integral of $\sin u$ is $-\cos u$ :

\frac{1}{3} \int \sin u \, du = \frac{1}{3} (-\cos u) + C

= -\frac{1}{3} \cos u + C

Step 6: Substitute back $x^3$ for $u$

Finally, replace $u$ with $x^3$ to get the answer in terms of $x$ :

\int x^2 \sin x^3 \, dx = -\frac{1}{3} \cos x^3 + C

Final Answer:

\int x^2 \sin x^3 \, dx = -\frac{1}{3} \cos x^3 + C

Choose the correct answer:

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