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

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

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

\int sin (x) cos (x) e^{s i n^{2} (x) + π} d x

\frac{1}{2} e^{π + s i n^{2} (x)} + C

Solution steps

\int sin (x) cos (x) e^{s i n^{2} (x) + π} d x

sin (x) cos (x) e^{s i n^{2} (x) + π} = e^{π} sin (x) cos (x) e^{s i n^{2} (x)}

= \int e^{π} sin (x) cos (x) e^{s i n^{2} (x)} d x

Take the constant out:

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

= e^{π} \cdot \int sin (x) cos (x) e^{s i n^{2} (x)} d x

Apply u-substitution

: \int e^{u^{2}} u d u

= e^{π} \cdot \int e^{u^{2}} u d u

Apply u-substitution

: \int \frac{e ^{v}}{2} d v

= e^{π} \cdot \int \frac{e ^{v}}{2} d v

Take the constant out:

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

= e^{π} \frac{1}{2} \cdot \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= e^{π} \frac{1}{2} e^{v}

Substitute back

= e^{π} \frac{1}{2} e^{s i n^{2} (x)}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{π} e^{s i n^{2} (x)} = e^{π + s i n^{2} (x)}

= \frac{1}{2} e^{π + s i n^{2} (x)}

Add a constant to the solution

= \frac{1}{2} e^{π + s i n^{2} (x)} + C

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