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

The integral of-sin(x)cos^7(x) is 1/8 cos^8(x)+C

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integral of-sin(x)cos^7(x)

\int - sin (x) cos^{7} (x) d x

\frac{1}{8} cos^{8} (x) + C

Solution steps

\int - sin (x) cos^{7} (x) d x

Take the constant out:

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

= - \int sin (x) cos^{7} (x) d x

Apply u-substitution

: \int - u^{7} d u

= - \int - u^{7} d u

Take the constant out:

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

= - (- \int u^{7} d u)

Apply the Power Rule

: \frac{u ^{8}}{8}

= - (- \frac{u ^{8}}{8})

Substitute back

u = cos (x)

= - (- \frac{cos ^{8} ( x )}{8})

Simplify

- (- \frac{cos ^{8} ( x )}{8}) : \frac{1}{8} cos^{8} (x)

= \frac{1}{8} cos^{8} (x)

Add a constant to the solution

= \frac{1}{8} cos^{8} (x) + C

\frac{d y}{d x} = \frac{( x - 4 )}{( y + 3 )}

x \to 1 + lim (x - 1)

\int 4 + 6 x + 36 x^{2} d x

(\frac{( 10 - x )}{4})^{2} + \frac{x ^{2}}{4 π}

What is the integral of-sin(x)cos^7(x) ?
The integral of-sin(x)cos^7(x) is 1/8 cos^8(x)+C