Algebra problem in integration by parts

Question

The integral to solve:

$$ \int{5^{sin(x)}cos(x)dx} $$

I used long computations using integration by parts, but I don't could finalize:

$$ \int{5^{sin(x)}cos(x)dx} = cos(x)\frac{5^{sin(x)}}{ln(5)}+\frac{1}{ln(5)}\Bigg[ \frac{5^{sin(x)}}{ln(5)}-\frac{1}{ln(5)}\int{5^{sin(x)}cos(x)dx} \Bigg] $$

Answer 1

If you do $\sin x=u$ and $\cos x\,\mathrm dx=\mathrm du$, your integral becomes$$\int 5^u\,\mathrm du.$$

Answer 2

Let $5^{\sin x}=y,$

As $5=e^{\ln5},$

$dy=5^{\sin x}\cos x\ln 5\ dx$ right?

Answer 3

$$\int{5^{sin(x)}cos(x)dx} = \cos(x)\frac{5^{sin(x)}}{ln(5)}+.........$$

You made a confusion with:

$$\int a^xdx =\int e^{x \ln |a|}dx=\dfrac {a^x}{\ln |a|}$$ You don't have $x$ here but a function of $x$ that is $\sin x$ Try by substitution $u=\sin x \implies du=\cos(x) dx$. Then you can apply the formula for $\int a^xdx$ : $$ I=\int 5^udu= \dfrac {5^u}{\ln 5}= \dfrac {5^{\sin x}}{\ln 5}$$