# Integral puzzles with solutions

My source is usually integration puzzles on exams and in maths societies.

# 1. $∫_{0}(tanx)_{31}dx$

## Answer

And

So

# 2. $∫_{0}cos(x)cos(a−x)xdx $

## Answer

Note that

and

Since the function is odd around $a/2$.

# 3. $∫exp(arccosx)dx$

## Answer

By substituting $x=cosθ$ and noting:

# 4. $∫_{0}1−x_{3}x dx$

## Answer

Classic substitution $x_{3}=u_{2}$, so $x_{21}dx=32 du$.

⚠ Might have got fractions the wrong way round ⚠

# 5. $∫_{0}1+x_{31}x_{−21} dx$

## Answer

Let $y_{6}=x$, then $dx=6y_{5}dy$. Let $T:=t_{61}$.