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Integrate the function w.r.t. $x$: $2x \sin (x^2 + 1)$ |
$-\cos (x^2 + 1) + C$ $\cos (x^2 + 1) + C$ $2\cos (x^2 + 1) + C$ $-\frac{1}{2}\cos (x^2 + 1) + C$ |
$-\cos (x^2 + 1) + C$ |
The correct answer is Option (1) → $-\cos (x^2 + 1) + C$ Derivative of $x^2 + 1$ is $2x$. Thus, we use the substitution $x^2 + 1 = t$ so that $2x \, dx = dt$. Therefore, $\int 2x \sin (x^2 + 1) \, dx = \int \sin t \, dt = -\cos t + C = -\cos (x^2 + 1) + C$ |
