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

The integral of sin(x)(cos(x)+7)^4 is -(cos^5(x))/5-7cos^4(x)-98cos^3(x)+686sin^2(x)-2401cos(x)+C

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integral of sin(x)(cos(x)+7)^4

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

- \frac{cos ^{5} ( x )}{5} - 7 cos^{4} (x) - 98 cos^{3} (x) + 686 sin^{2} (x) - 2401 cos (x) + C

Solution steps

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

Expand

sin (x) (cos (x) + 7)^{4} : cos^{4} (x) sin (x) + 28 cos^{3} (x) sin (x) + 294 cos^{2} (x) sin (x) + 1372 sin (x) cos (x) + 2401 sin (x)

= \int cos^{4} (x) sin (x) + 28 cos^{3} (x) sin (x) + 294 cos^{2} (x) sin (x) + 1372 sin (x) cos (x) + 2401 sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{4} (x) sin (x) d x + \int 28 cos^{3} (x) sin (x) d x + \int 294 cos^{2} (x) sin (x) d x + \int 1372 sin (x) cos (x) d x + \int 2401 sin (x) d x

\int cos^{4} (x) sin (x) d x = - \frac{cos ^{5} ( x )}{5}

\int 28 cos^{3} (x) sin (x) d x = - 7 cos^{4} (x)

\int 294 cos^{2} (x) sin (x) d x = - 98 cos^{3} (x)

\int 1372 sin (x) cos (x) d x = 686 sin^{2} (x)

\int 2401 sin (x) d x = - 2401 cos (x)

= - \frac{cos ^{5} ( x )}{5} - 7 cos^{4} (x) - 98 cos^{3} (x) + 686 sin^{2} (x) - 2401 cos (x)

Add a constant to the solution

= - \frac{cos ^{5} ( x )}{5} - 7 cos^{4} (x) - 98 cos^{3} (x) + 686 sin^{2} (x) - 2401 cos (x) + C

\int \frac{e ^{x}}{1 + ( e ^{x} ) ^{2}} d x

\int (x + 8) 16 x + x^{2} d x

\int (5^{x} - 3^{x}) d x

\int 12 x^{- 3} d x

What is the integral of sin(x)(cos(x)+7)^4 ?
The integral of sin(x)(cos(x)+7)^4 is -(cos^5(x))/5-7cos^4(x)-98cos^3(x)+686sin^2(x)-2401cos(x)+C