Question

The value of integral xcosx^2 dx is​

Answer 1

EXPLANATION.

⇒ ∫x. cos(x²)dx.

As we know that,

By using substitution method in this equation, we get.

Let we assume that,

⇒ x² = t.

Differentiate w.r.t x, we get.

⇒ 2xdx = dt.

⇒ x dx = dt/2.

Put the value in the equation, we get.

⇒ ∫cos(t)dt/2.

⇒ 1/2∫cos(t)dt.

⇒ 1/2 sin(t) + c.

Put the value of t = x² in the equation, we get.

⇒ 1/2 sin(x²) + c.

                                                                                                                         

MORE INFORMATION.

Integration using substitution.

We have two more substitution techniques for particular type of integration.

(1) = ∫f'(x)/f(x) = ㏒ | f(x) | + c.

Proof :

Let we assume that,

⇒ f(x) = t.

Differentiate w.r.t t, we get.

⇒ dt/dx = f'(x).

⇒ dt = f'(x)dx.

Put the value in the equation, we get.

⇒ ∫dt/t.

⇒ ㏒ |t| + c.

Put the value of t = f(x) in the equation, we get.

⇒ ㏒ |f(x)| + c.

(2) = ∫f'(x). (f(x))ⁿdx = f(x)ⁿ⁺¹/n + 1 + c.

Proof :

Let we assume that,

⇒ f(x) = t.

⇒ dt/dx = f'(x).

⇒ dt = f'(x)dx.

⇒ ∫tⁿdt.

⇒ tⁿ⁺¹/n + 1 + c.

⇒ (f(x))ⁿ⁺¹/n + 1 + c.