What is the integral of x^7cos(x^4) ?

The integral of x^7cos(x^4) is 1/4 (x^4sin(x^4)+cos(x^4))+C

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integral of x^7cos(x^4)

\int x^{7} cos (x^{4}) d x

\frac{1}{4} (x^{4} sin (x^{4}) + cos (x^{4})) + C

Solution steps

\int x^{7} cos (x^{4}) d x

Apply u-substitution

: \int \frac{u cos ( u )}{4} d u

= \int \frac{u cos ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{4} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{4} (u sin (u) - (- cos (u)))

Substitute back

u = x^{4}

= \frac{1}{4} (x^{4} sin (x^{4}) - (- cos (x^{4})))

Simplify

= \frac{1}{4} (x^{4} sin (x^{4}) + cos (x^{4}))

Add a constant to the solution

= \frac{1}{4} (x^{4} sin (x^{4}) + cos (x^{4})) + C

\int_{- 1}^{- \frac{1}{2}} - cos (kπ x) d x

\int sin (x) - 1 d x

\frac{s + 1}{s ^{2} ( s + 2 )}

\frac{d ^{2} y}{d x ^{2}} = y

\int - cos (x) ln (x) d x

What is the integral of x^7cos(x^4) ?
The integral of x^7cos(x^4) is 1/4 (x^4sin(x^4)+cos(x^4))+C