What is the integral from 0 to x^{10 of}cos(sqrt(t)) ?

The integral from 0 to x^{10 of}cos(sqrt(t)) is 2(x^5sin(x^5)+cos(x^5)-1)

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integral from 0 to x^{10 of}cos(sqrt(t))

\int_{0}^{x^{10}} cos (t) d t

2 (x^{5} sin (x^{5}) + cos (x^{5}) - 1)

Solution steps

\int_{0}^{x^{10}} cos (t) d t

Apply u-substitution

: \int_{0}^{x^{10}} cos (u) \cdot 2 u d u

= \int_{0}^{x^{10}} cos (u) \cdot 2 u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{x^{10}} cos (u) u d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{x^{10}}

= 2 [u sin (u) - \int sin (u) d u]_{0}^{x^{10}}

\int sin (u) d u = - cos (u)

= 2 [u sin (u) - (- cos (u))]_{0}^{x^{10}}

Simplify

2 [u sin (u) - (- cos (u))]_{0}^{x^{10}} : 2 [u sin (u) + cos (u)]_{0}^{x^{5}}

= 2 [u sin (u) + cos (u)]_{0}^{x^{5}}

Compute the boundaries:

x^{5} sin (x^{5}) + cos (x^{5}) - 1

= 2 (x^{5} sin (x^{5}) + cos (x^{5}) - 1)

\int_{0}^{\frac{π}{3}} \frac{1}{cos ^{2} ( 2 x )} d x

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\int_{0}^{2 π} 1 + t^{2} d t

\int_{0}^{\frac{π}{6}} cos (2 t) d t

\int_{2}^{0} c x d x

What is the integral from 0 to x^{10 of}cos(sqrt(t)) ?
The integral from 0 to x^{10 of}cos(sqrt(t)) is 2(x^5sin(x^5)+cos(x^5)-1)