What is the integral of sin(2x)(e^{1+cos^2(x)})+3 ?

The integral of sin(2x)(e^{1+cos^2(x)})+3 is -e^{1+cos^2(x)}+3x+C

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integral of sin(2x)(e^{1+cos^2(x)})+3

\int sin (2 x) (e^{1 + c o s^{2} (x)}) + 3 d x

- e^{1 + c o s^{2} (x)} + 3 x + C

Solution steps

\int sin (2 x) e^{1 + c o s^{2} (x)} + 3 d x

Simplify

e^{1 + c o s^{2} (x)} : e e^{c o s^{2} (x)}

= \int sin (2 x) e e^{c o s^{2} (x)} + 3 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 sin (2 x) e e^{c o s^{2} (x)} d x + \int 3 d x

\int sin (2 x) e e^{c o s^{2} (x)} d x = - e^{1 + c o s^{2} (x)}

\int 3 d x = 3 x

= - e^{1 + c o s^{2} (x)} + 3 x

Add a constant to the solution

= - e^{1 + c o s^{2} (x)} + 3 x + C

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

\int sin (\frac{1}{x}) d x

f (x) = cot^{3} (x)

What is the integral of sin(2x)(e^{1+cos^2(x)})+3 ?
The integral of sin(2x)(e^{1+cos^2(x)})+3 is -e^{1+cos^2(x)}+3x+C