What is the integral of 10x^9cos(x^{10}) ?

The integral of 10x^9cos(x^{10}) is sin(x^{10})+C

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integral of 10x^9cos(x^{10})

\int 10 x^{9} cos (x^{10}) d x

sin (x^{10}) + C

Solution steps

\int 10 x^{9} cos (x^{10}) d x

Take the constant out:

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

= 10 \cdot \int x^{9} cos (x^{10}) d x

Apply u-substitution

: \int \frac{cos ( u )}{10} d u

= 10 \cdot \int \frac{cos ( u )}{10} d u

Take the constant out:

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

= 10 \cdot \frac{1}{10} \cdot \int cos (u) d u

Use the common integral:

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

= 10 \cdot \frac{1}{10} sin (u)

Substitute back

u = x^{10}

= 10 \cdot \frac{1}{10} sin (x^{10})

Simplify

10 \cdot \frac{1}{10} sin (x^{10}) : sin (x^{10})

= sin (x^{10})

Add a constant to the solution

= sin (x^{10}) + C

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What is the integral of 10x^9cos(x^{10}) ?
The integral of 10x^9cos(x^{10}) is sin(x^{10})+C