Question

∫xcosax^2dx
with respect to x

cos^3x+c
sin2x+c
sec^2x+1
1/2asinax^2+c

Answer 1

Answer on Question #46134– Math – Integral Calculus

Question:

Integrate the following expression with respect to $x$:

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

cos^3x+c

sin2x+c

sec^2x+1

1/2asinax^2+c

Solution:

Let us change the variable of integration

$$t = a x ^ {2}.$$

Then the differential $dx$ takes the form

$$d x = d \sqrt {\frac {t}{a}} = \frac {d t}{2 \sqrt {t a}}.$$

Therefore

$$\int x \cdot \cos a x ^ {2} \cdot d x = \int \sqrt {\frac {t}{a}} \cdot \cos t \cdot \frac {d t}{2 \sqrt {t a}} = \frac {1}{2 a} \int \cos t \cdot d t.$$

Consequently

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

Thus the final answer is

$$\int x \cdot \cos a x ^ {2} \cdot d x = \frac {\sin a x ^ {2}}{2 a} + \text{const}.$$

Answer: the last answer is correct

$$\int x \cdot \cos a x ^ {2} \cdot d x = \frac {\sin a x ^ {2}}{2 a} + \text{const}.$$

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