Question

Find the integral of x cos ax^2 dx with respect to x

a. cos^3x + c

b. sin^2x + c

c. sec^2x + 1

d. 1/2 a sin ax^2 + c

Answer 1

Find the integral of $x \cos ax^2 dx$ with respect to $x$

a. $\cos^3 x + c$

b. $\sin^2 x + c$

c. $\sec^2 x + 1$

d. $\frac{1}{2a} \sin ax^2 + c$

**Solution.**

Let's integrate it:

$$\int x \cos a x ^ {2} d x$$

Substitute $a x^{2} = t$, then $dt = 2a x d x$. We have

$$\int x \cos a x ^ {2} d x = \int \frac {1}{a ^ {2}} \cos a x ^ {2} \cdot 2 a x d x = \frac {1}{2 a} \int \cos t d t = \frac {1}{2 a} \sin t + c$$

Substitute back $t = a x^{2}$:

$${\frac {1}{2 a}} \sin t + c = {\frac {1}{2 a}} \sin a x ^ {2} + c$$

**Answer:**

d. $\frac{1}{2a} \sin ax^2 + c$