Use skills to solve this integral

Question

How to use technique to integrate this integral? $\int_{0}^{\frac{\pi}{4}} \cos t\cdot\sin t e^{\sin t}-\sin^2t \cdot \tan te^{\cos t} +\tan te^{\tan t} dt$

I use integration by parts for many many times.

Answer 1

Let us consider the three antiderivatives $$I_1=\int \cos (t)\sin (t)\,e^{\sin (t)} dt$$ Let $\sin(t)=x$ to arrive at $$I_1=\int x e^x \,dx= (x-1)e^x= (\sin (t)-1)\,e^{\sin (t)}$$ $$I_2=\int \sin ^2(t) \tan (t)\,e^{\cos (t)}\,dt$$ Let $\cos(t)=x$ to arrive at $$I_2=\int \frac{e^x \left(x^2-1\right)}{x}\,dx=\int x e^x \,dx-\int\frac{e^x} x \,dx=(x-1)e^x-\int\frac{e^x} x \,dx$$ that is to say $$I_2=(\cos (t)-1)\,e^{\cos (t)}-\int \frac {e^{\cos(t)}}{\cos(t)} dt$$ $$I_3=\int \tan (t)\, e^{\tan (t)}\,dt$$ Let $\tan(t)=x$ to arrive at $$I_3=\int\frac{x\,e^x }{x^2+1}\,dx=\frac{1-i}2\left(\int \frac{e^x}{x+i}\,dx-\int \frac{e^x}{x-i}\,dx \right)$$ All the unsolved integrals are very simple if you are familiar with the exponential integral function.

If you are not, compute $$I_3=\int \tan (t)\, e^{\tan (t)}\,dt \qquad \text{and} \qquad I_4=\int \frac {e^{\cos(t)}}{\cos(t)} dt $$ using series expansions and integrate termwise.

Finsih using the bounds.