How do you think there can be integration when almost 1/3 of the inhabitants of a city are immigrants?

Not by alienating them but by teaching them and their children your values and customs. If you are confident that your values are universal and coherent, they will propagate with little friction. Toulouse and Montpellier for instance have way more than a third of immigrants and yet they are not ghettos or areas with cultural conflicts as pronounced as in the UK.

I have heard that France also has big problems with certain cities and places with a high percentage of immigrants. Isn't Marseille the crime capital of France? Or the Banlieus around Paris?

Marseille is the headquarters of the French mafia, there have always been more corruption, crime, weapon availability there, just like in Sicily or Milan. This network is finishing to fall and of course others have taken the place. Most drugs will transit via Spain or the Mediterranean because of the places where it is produced, thus you won't find as much organised crime in Western France where the drug that transits is for local rather than national distribution. There are now many Eastern European and Chinese criminal networks (prostitution especially), and this is where all the weapons used by gangs come from. Yet there is no city that has a third of Eastern Europeans or Chinese.

If you want to ignore the historical location argument, compare Marseille, ruled by a despot that has always done his best to isolate the poor community in one section of the city, with Montpellier or Toulouse which have very large migrant populations but socialist rulers in the past few decades. Same local culture, same proportion of immigrants, but the crime levels are completely different. Paris Suburbs were built in the 60's in a completely nonsensical way and people were packed there (not just Muslims, either; lots of Poles and Portuguese too, lots of Vietnamese, etc.). People who arrived at the same time, were put in a city that was artificially rather than organically built, and who ran out of work as soon as the post-war reconstruction was done. There is no need to look for ethnic or religious arguments here. The same ghettoisation happens in Lille (former French mining industry) or Manchester in the UK, or in cities in the US where the car industry was located. Rapid growth of unemployment in a specific area undermines it for decades, regardless of who lives there.

Of course we have issues with ethnic and religious groups in France. Tons of groups. There is a close Chinese community made of recent immigrants that is absolutely not integrated compared to the third-generation Viet community. There are many small religious and ethnic communities in big cities, and some specific areas that have odd ethnic groups (like the Harkis in Montpellier). But those issues are nothing compared to the social and economic causes of ghetto expansion. They're barely even worth mentioning because they're a non-problem. Most of the mass immigration of the past 50 years is overall well integrated culturally, even though it is not economically. There is no global immigrant problem in France, or elsewhere as far as I'm concerned.