Integral of $2 x \cos\left(x^{2}\right)$

Q: What is the integral of 2x * cos(x^2)?

The integral of 2 x ≤ft(x^2) is ≤ft(x^2) + C.

The integral of $2 x \cos\left(x^{2}\right)$ is $\sin\left(x^{2}\right) + C$ .

\int 2 x \cos\left(x^{2}\right)\,dx = \sin\left(x^{2}\right) + C

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Graph of f(x) = 2x * cos(x^2) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.84147098.

How to work it out

1
We want an antiderivative of f(x) = 2x * cos( $x^{2}$ ).
$\int 2 x \cos\left(x^{2}\right)\,dx$
2
Method: substitution u = x².
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int 2 x \cos\left(x^{2}\right)\,dx = \sin\left(x^{2}\right) + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\sin\left(x^{2}\right)\right] = \cos\left(x^{2}\right) 2 x$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} 2 x \cos\left(x^{2}\right)\,dx = \Big[\sin\left(x^{2}\right)\Big]_{0}^{1} \approx 0.84147098$

LaTeX source of the formulas on this page

\[ \int 2 x \cos\left(x^{2}\right)\,dx = \sin\left(x^{2}\right) + C \]
\[ \int 2 x \cos\left(x^{2}\right)\,dx \]
\[ \frac{d}{dx}\left[\sin\left(x^{2}\right)\right] = \cos\left(x^{2}\right) 2 x \]
\[ \int_{0}^{1} 2 x \cos\left(x^{2}\right)\,dx = \Big[\sin\left(x^{2}\right)\Big]_{0}^{1} \approx 0.84147098 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.84147098
from 0 to 0.5	0.24740396

Questions

What is the integral of 2x * cos(

x^{2}

The integral of $2 x \cos\left(x^{2}\right)$ is $\sin\left(x^{2}\right) + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of 2x * cos(

x^{2}

) from 0 to 1?

About 0.84147098.

Integral of 2xcos⁡(x2)2 x \cos\left(x^{2}\right)2xcos(x2)

How to work it out

Definite integrals of the same function

Questions

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Integral of $2 x \cos\left(x^{2}\right)$